Methods for propagating a non sinusoidal signal without distortion in dispersive lossy media

ABSTRACT

Systems and methods are described for transmitting a waveform having a controllable attenuation and propagation velocity. An exemplary method comprises: generating an exponential waveform, the exponential waveform (a) being characterized by the equation V in =De −A     SD     [x−v     SD     t] , where D is a magnitude, V in  is a voltage, t is time, A SD  is an attenuation coefficient, and v SD  is a propagation velocity; and (b) being truncated at a maximum value. An exemplary apparatus comprises: an exponential waveform generator; an input recorder coupled to an output of the exponential waveform generator; a transmission line under test coupled to the output of the exponential waveform generator; an output recorder coupled to the transmission line under test; an additional transmission line coupled to the transmission line under test; and a termination impedance coupled to the additional transmission line and to a ground.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of, and claims priority to, U.S.patent application Ser. No. 11/010,198, filed on Dec. 10 2004, which nowU.S. Pat. No. 7,375,602 is a continuation-in-part of U.S. patentapplication Ser. No. 10/224,541 filed Aug. 20, 2002, now U.S. Pat. No.6,847,267, which is a continuation-in-part of U.S. patent applicationSer. No. 09/519,922, filed Mar. 7, 2000, now U.S. Pat. No. 6,441,695.The entire text of each of the above related applications isincorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to the field of signal transmission.More particularly, a representative embodiment of the invention relatesto transmission line testing and communication.

2. Discussion of the Related Art

Transmission lines, with their characteristic loss of signal as well asinherent time delay, may create problems in designing systems thatemploy a plurality of signals, which may be subject to delay anddistortion. Typical signals used to generate inputs to transmissionlines generally exhibit delay or propagation times that are not easilydeterminable. The propagation velocity of these waves is also variablewith displacement along the transmission line.

A Time Domain Reflectometer (TDR) is a test instrument used to findfaults in transmission lines and to empirically estimate transmissionline lengths and other parameters characterizing the line, such as:inductance per unit length, capacitance per unit length, resistance perunit length and conductance per unit length.

An important measurement in TDR test technology is the time-of-flight(TOF) of a test pulse generated by the instrument and applied to theline. The time-of-flight may be measured by timing the passage of thepulse detected at two locations along the line. Along with a value ofthe propagation speed of the pulse, time-of-flight measurements canallow one to obtain the distance between measurement points or, in thecase of a reflected wave, the distance from the pulse launch point tothe location of the impedance change causing the pulse to be reflectedand returned.

A fundamental limitation in TDR technology is the poor accuracy of TOFmeasurements in lossy, dispersive transmission lines. The relativelyhigh TDR accuracy of TOF values obtainable in short low loss, lowdispersion transmissions lines is possible only because the propagatingtest pulses keep their shape and amplitude in tact over the distancesthey travel during TOF measurements. By contrast, in dispersive, lossylong transmission lines the test pulses used in the art change shape,amplitude, and speed as they travel.

Further, it is difficult to provide high-speed communications in alossy, frequency dependent transmission media. It would be advantageousto have a method to increase data transmission rates in suchtransmission lines.

Until now, the requirements of providing a method and/or apparatus foraccurately measuring times-of-flight, estimating line lengths and otherparameters characterizing lossy, dispersive transmission lines, andproviding high-speed communications via such transmission media have notbeen met. What is needed is a solution that addresses theserequirements.

SUMMARY OF THE INVENTION

Shortcomings are reduced or eliminated by the techniques disclosed here.In one respect, the disclosure involves a method for transmitting anexponential waveform. The exponential waveform, which can becharacterized by the equation V_(SD)=De^(−A) ^(SD) ^([x−v) ^(SD) ^(t])and can be truncated at a maximum value and can have an essentiallyconstant shape during transmission on a transmission line.

In another respect, the disclosure involves transmitting a waveformincluding a speedy delivery signal envelope modulated with a sinusoidalsignal on a media. The waveform can be characterize by the equationV(z,s)=B(s)e^(−zγ(s)) and having a substantially constant shape duringtransmission on the media. In some embodiments, an exponential waveformcan include a truncated speedy delivery signal envelope modulated with asinusoidal carrier signal, in which the media can be aresistance-capacitance-inductance transmission line. In otherembodiments, an electromagnetic plane waveform can include a truncatedspeedy delivery signal envelope modulated with a sinusoidalelectromagnetic plane wave on a media, such as a dispersive plasmamedia.

In other respects, the disclosure involves a method for determining atemperature of a media. The method includes generating an exponentialwaveform on the media and determining the delay of the exponentialwaveform. The delay can be characterized by the equation

$\delta_{SD} = {\left( \frac{1}{\alpha} \right) \cdot \sqrt{{{\overset{\_}{L}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}\alpha^{2}} + {\begin{pmatrix}{{{\overset{\_}{G}\left( {\alpha,T} \right)}{\overset{\_}{L}\left( {\alpha,T} \right)}} +} \\{{\overset{\_}{R}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}}\end{pmatrix}\alpha} + {{\overset{\_}{R}\left( {\alpha,T} \right)}{\overset{\_}{G}\left( {\alpha,T} \right)}}}}$Where the temperature is proportional to the length of travel of theexponential waveform on the media.

In another respect, a method is provided for measuring a temperature ofa media. The method includes generating a speedy delivery (SD) signal ona media and determining the delay of the SD signal characterized by theequation Δ_(SD)=l(T)·√{square root over (L(T)C(T))}{square root over(L(T)C(T))}·√{square root over (1+τ/(L(T)/R(T)))}{square root over(1+τ/(L(T)/R(T)))}. The method also includes determining the temperatureof the media which can be proportional to the delay.

In yet another respect, the disclosure involves a method for reducinginterconnect delays. A signal is generated on an interconnect and thedelay of the signal can be characterized by a line model and theequation δ_(SD)=√{square root over (LC)}√{square root over (1+τ/(L/R))}.In one embodiment, the method includes reducing the delay by decreasingτ. In another embodiment, the method includes inserting a repeater onthe interconnect.

In other respects, the disclosure involves a method including steps forgenerating an exponential waveform having an essentially constant shapeduring transmission on a media at a first location. A plurality ofexponentials can be encoded on a leading edge of the exponentialwaveform, where the encoded exponential waveform is transmitted to asecond location. The method also includes decoding the encodedexponential waveform at the second location.

The disclosure also involves a method for determining a length of aninterconnect for a desired delay. A signal can be generated and a lengthof the interconnect can be characterized by the equation

${l = \frac{\Delta_{SD}}{\sqrt{LC} \cdot \sqrt{1 + \frac{\tau}{L/R}}}},$where L is the inductance per unit length of the interconnect, Rresistance per unit length of the interconnect, C is the capacitance perunit length of the interconnect, and Δ_(SD) is the desired delay.

A thermometer is also presented. The thermometer includes a signalgenerator configured to generate a truncated exponential waveformcharacterized by the equation V_(SD)=De^(−A) ^(SD) ^([x−v) ^(SD) ^(t])where D is a magnitude, V_(SD) is a voltage, t is time, A_(SD) is anattenuation coefficient, v_(SD) is a propagation velocity, and t istime. The thermometer also includes a processor coupled to the signalgenerator. The processor is configured to determine a temperature of themedia, where the temperature of the media being proportional to thedelay of the waveform.

In another respect, the present disclosure includes a time domainreflectometer. The time domain reflectometer includes a signal generatorconfigured to generate a truncated exponential waveform on a media.Using the generated truncated exponential media, the time domainreflectometer is configured to determine the length of the media. Inanother embodiment, the time domain reflectometer is configured todetect the locations of possible faults.

These, and other, embodiments of the invention will be betterappreciated and understood when considered in conjunction with thefollowing description and the accompanying drawings. It should beunderstood, however, that the following description, while indicatingvarious embodiments of the invention and numerous specific detailsthereof, is given by way of illustration and not of limitation. Manysubstitutions, modifications, additions and/or rearrangements may bemade within the scope of the invention without departing from the spiritthereof, and the invention includes all such substitutions,modifications, additions and/or rearrangements.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings accompanying and forming part of this specification areincluded to depict certain aspects of the invention. A clearerconception of the invention, and of the components and operation ofsystems provided with the invention, will become more readily apparentby referring to the exemplary, and therefore nonlimiting, embodimentsillustrated in the drawings, wherein like reference numerals (if theyoccur in more than one view) designate the same or similar elements. Theinvention may be better understood by reference to one or more of thesedrawings in combination with the description presented herein. It shouldbe noted that the features illustrated in the drawings are notnecessarily drawn to scale.

FIG. 1A is a graph of an SD waveform for (x=0,t), representing anembodiment of the invention.

FIG. 1B is a graph of an SD waveform in moving frame at x′=l (x=l,t′),representing an embodiment of the invention.

FIG. 2 is block diagram of a setup to measure SD parameters,representing an embodiment of the invention.

FIG. 3 is a graph of voltage traces at x=0 and x=1002 ft of a 2004 fttransmission line (T1 cable) with α=3.0×10⁶ l/sec indicatingtime-of-flight measured at 4V threshold, representing an embodiment ofthe invention.

FIG. 4A is a graph of SD threshold overlap regions for voltages at x=0and at x=d (1002 ft), representing an embodiment of the invention.

FIG. 4B is a graph of measured constant threshold times-of-flight for1002 ft T1 line with α=3.0×10⁶ l/sec, representing an embodiment of theinvention.

FIG. 5A is a graph of SD threshold overlap regions for voltages at x=0and ½ voltage across R_(t)=∞, representing an embodiment of theinvention.

FIG. 5B is a graph of measured constant threshold times-of-flight for2004 ft T1 line with α=3.0×10⁶ l/sec, representing an embodiment of theinvention.

FIG. 6A is a graph of an input pulse applied to a T1 cable toempirically determine a transfer function, representing an embodiment ofthe invention.

FIG. 6B is a graph of a pulse measured at 1002 ft for a T1 cable,representing an embodiment of the invention.

FIG. 7A is a graph of a Power Spectral Density of an input pulse appliedto a T1 cable, representing an embodiment of the invention.

FIG. 7B is a graph of a Power Spectral Density of a pulse measured at1002 ft for a T1 cable, representing an embodiment of the invention.

FIG. 8 is a graph of an empirically derived transfer function for a 1002ft T1 cable, representing an embodiment of the invention.

FIG. 9 is a graph of a measured initial waveform with simulated 1002 ftwaveform for T1 cable and constant threshold times-of-flight,representing an embodiment of the invention.

FIG. 10 is a graph of a difference between measured and simulated 1002ft voltage traces, representing an embodiment of the invention.

FIG. 11 is a graph of a simulated initial waveform with 1002 ft waveformfor T1 cable, α=1×10⁵ sec⁻¹, and constant threshold times-of-flight,representing an embodiment of the invention.

FIG. 12 is a graph of a simulated initial waveform with 1002 ft waveformfor T1 cable, α=1×10⁷ sec⁻¹, and constant threshold times-of-flight,representing an embodiment of the invention.

FIG. 13A is a graph of SD parameters V_(SD) determined by simulationwith empirical transfer function, representing an embodiment of theinvention.

FIG. 13B is a graph of SD parameters A_(SD) determined by simulationwith empirical transfer function, representing an embodiment of theinvention.

FIG. 14 is block diagram of a setup to determine Z_(SD)(α), representingan embodiment of the invention.

FIG. 15A is a graph of voltage traces measured at d=1002 ft with varyingtermination resistances, α=1.5×10⁶ l/sec, representing an embodiment ofthe invention.

FIG. 15B is a graph of ratios of output signal amplitudes with knowntermination traces and one-half of line output signal amplitude for opentermination trace, α=1.5×10⁶ l/sec, representing an embodiment of theinvention.

FIG. 16 is a graph of experimentally determined SD impedances for a T1cable as a function of α, representing an embodiment of the invention.

FIG. 17 is a graph of SD voltage traces and the natural logarithm of SDvoltage traces measured at input, ˜6 kft, ˜12 kft and ˜18 kft along 24AWG twisted wire pairs connected in series, representing an embodimentof the invention.

FIG. 18 is a graph used to detect end of SD region demonstratingsensitivity to noise and speed of response for the filters at 12 kft.,representing an embodiment of the invention.

FIG. 19 is a graph of a variation of 1/√{square root over (LC)} as afraction of the speed of light in a vacuum with α, representing anembodiment of the invention.

FIG. 20 is a graph of an input waveform attenuated and time shifted for11,573 ft (Top), 12,219 ft (Middle) and 12,859 ft (Bottom) shown withwaveform measured at ˜12 kft, representing an embodiment of theinvention.

FIG. 21 is a graph of a standard deviation of V_(Predicted)−V_(Measured)in the SD region vs. estimated distance in region of correlation,representing an embodiment of the invention.

FIG. 22A is a graph of V_(Predicted)−V_(Measured) and minimum varianceof V_(Predicted)−V_(Measured) at estimated distance of 12,216 ft,representing an embodiment of the invention.

FIG. 22B is a graph of V_(Predicted)−V_(Measured) and variance ofV_(Predicted)−V_(Measured) at estimated distance of 12,254 ft,representing an embodiment of the invention.

FIG. 23 is a graph of voltage traces measured a ˜6, ˜12, and ˜18 kft of24 AWG BKMA 50 cable and applied waveform attenuated and shifted in timeaccording to SD prediction, representing an embodiment of the invention.

FIG. 24 is a graph of voltage trace measured at input showing appliedtruncated SD pulse and wave reflected from open termination at 1002 ft,representing an embodiment of the invention.

FIG. 25 is a graph of voltage trace measured at input showing wavereflected from open termination at 1002 ft and applied wave attenuatedand shifted forward in time for best fit resulting in 2004 ft estimatedtotal distance traveled, representing an embodiment of the invention.

FIG. 26A is a graph of SD TDR threshold overlap regions for voltages atx=0 for 1002 ft open terminated T1 line, representing an embodiment ofthe invention.

FIG. 26B is a graph of measured constant threshold TDR times-of-flightfor 1002 ft T1 line with α=6.7×10⁶ l/sec resulting in total estimateddistance traveled by the reflected wave of 2010 ft., representing anembodiment of the invention.

FIG. 27 is a graph of voltage trace measured at input showing appliedtruncated SD pulse and wave reflected from short circuit termination at1002 ft., representing an embodiment of the invention.

FIG. 28 is a graph of a voltage trace measured at input inverted,showing wave reflected from short circuit termination at 1002 ftinverted and the non-inverted applied input wave attenuated and shiftedforward in time for best fit resulting in 2004 ft estimated totaldistance traveled, representing an embodiment of the invention.

FIG. 29A is a graph of SD TDR threshold overlap regions for voltages atx=0 for 1002 ft short circuit terminated T1 line, representing anembodiment of the invention.

FIG. 29B is a graph of measured constant threshold TDR times-of-flightfor 1002 ft T1 line with α=6.7×10⁶ l/sec resulting in total estimateddistance traveled of 2008 ft., representing an embodiment of theinvention.

FIG. 30 is a graph of a voltage pulse applied to 1002 ft T1 cable(dotted line), and to resistor with value Z_(o), representing anembodiment of the invention.

FIG. 31 is another graph of the voltage pulse applied to 1002 ft T1cable (dotted line), and to resistor with value Z_(o), representing anembodiment of the invention.

FIG. 32 is a graph of a transfer function between the difference ofwaveform applied to the twisted wire pair and the waveform applied toZ_(o), representing an embodiment of the invention.

FIG. 33 is a graph of simulated applied SD waveform with α=6.7×10⁶l/sec, and reflected waveform calculated with transfer function,representing an embodiment of the invention.

FIG. 34 is a graph of simulated voltage traces showing wave reflectedfrom open termination at 1002 ft and applied wave attenuated and shiftedforward in time for best fit resulting in 2004 ft estimated totaldistance traveled, representing an embodiment of the invention.

FIG. 35A is a graph of simulated SD TDR threshold overlap regions forvoltages at x=0 for 1002 ft open terminated T1 line, representing anembodiment of the invention.

FIG. 35B is a graph of simulated constant threshold TDR times-of-flightfor 1002 ft T1 line with α=6.7×10⁶ l/sec resulting in total estimateddistance traveled of 2005 ft, representing an embodiment of theinvention.

FIG. 36A is a graph of an uncompensated SD waveform applied to a 2004 fttwisted wire pair transmission line, representing an embodiment of theinvention.

FIG. 36B is a graph of an uncompensated SD waveform measured 100 Ohmtermination on a 2004-ft twisted wire pair transmission line,representing an embodiment of the invention.

FIG. 37A is a graph of a compensated SD waveform applied to a 2004 fttwisted wire pair transmission line, representing an embodiment of theinvention.

FIG. 37B is a graph of a compensated SD waveform measured 100 Ohmtermination on a 2004-ft twisted wire pair transmission line,representing an embodiment of the invention.

FIG. 38 is a graph of a series of four symbols measured at input to a2004 ft twisted wire pair transmission line, representing an embodimentof the invention.

FIG. 39 is a graph of a series of four symbols measured at 100 Ohmtermination at 2004-ft on twisted wire pair transmission line,representing an embodiment of the invention.

FIG. 40 is a natural logarithmic graph of voltages measured at 100 ohmtermination at 2004-ft for each symbol period showing a least meansquare fit of linear region, representing an embodiment of theinvention.

FIG. 41 is a graph of X_(i)=e^((α) ^(o) ^(+(i−1)·Δα)·t), representing anembodiment of the invention.

FIG. 42 is a graph of an orthonormal vector set Y=[Y₁, Y₂, Y₃, Y₄, Y₅],representing an embodiment of the invention.

FIG. 43 is a graph of non-orthogonal waveforms

${Y^{\prime} = {\begin{bmatrix}{Y_{1},} & {Y_{2},} & {Y_{3},} & {Y_{4},} & Y_{5}\end{bmatrix} \cdot {\sin\left( {\frac{\pi}{7 \times 10^{- 6}} \cdot t} \right)}}},$representing an embodiment of the invention.

FIG. 44 is a graph of dispersed waveforms Y′ at the receiver at 8000 ft,representing an embodiment of the invention.

FIG. 45 is a graph of a compensating Pulse C, at Input and at 8000 ft,representing an embodiment of the invention.

FIG. 46 is a graph of component functions at input with compensatingpulses, Z=Y′+b·C, representing an embodiment of the invention.

FIG. 47 is a graph of component functions Z at 8 kft, representing anembodiment of the invention.

FIG. 48 is a graph of linear independent waveforms, S, transmitted togenerate orthogonal waveforms at 8 kft, representing an embodiment ofthe invention.

FIG. 49 is a graph of orthogonal waveforms, S, received at 8 kft,representing an embodiment of the invention.

FIG. 50 is a flow diagram of an orthonormal set, S, generationalgorithm, representing an embodiment of the invention.

FIG. 51 is a graph of three successive transmitted symbols, Q, with fivebits encoded on each of four orthonormal pulses of S, representing anembodiment of the invention.

FIG. 52 is a graph of three successive symbols, Q, at the input to thereceiver at 8 kft with five bits encoded on each of four orthonormalpulses of S, representing an embodiment of the invention.

FIG. 53 is a graph of histogram of signal error,a_(expected)−a_(detected), for one second, or 1.67×10⁶ bits, of datatransmitted with −140 dBm AWGN present, representing an embodiment ofthe invention.

FIG. 54 is a block diagram of an apparatus, representing an embodimentof the invention.

FIG. 55 is a graph of signal delay measurements of a media at differenttemperatures, representing an embodiment of the invention.

FIG. 56 is a graph of average temperatures versus average signal delays,representing an embodiment of the invention.

FIG. 57 is a graph of line delays versus a signal waveform parameter,representing an embodiment of the invention.

FIG. 58 is a graph of a signal attenuation as a function of line length,representing an embodiment of the invention.

FIG. 59 is a graphical user interface showing two parameters encodedinto a leading edge of a signal, representing an embodiment of theinvention.

FIG. 60 are graphs of an encoded signal of FIG. 59 and a reflectedsignal of the encoded signal, representing an embodiment of theinvention.

FIG. 61 are natural log graphs of an exponential portion of a lowerportion of the leading edge on the encoded signal of FIG. 60,representing an embodiment of the invention.

FIG. 62 are graphs of the output of ratio of filters used on the encodedsignal of FIG. 60, representing an embodiment of the invention.

FIG. 63A are graphs of an encoded signal of FIG. 59 and a reflectedsignal of the encoded signal, representing an embodiment of theinvention.

FIG. 63B are natural log graphs of an exponential portion of an upperportion of the leading edge on the encoded signal of FIG. 63A,representing an embodiment of the invention.

FIG. 63C are graphs of the output of ratio of filters used on theencoded signal of FIG. 63A, representing an embodiment of the invention.

FIG. 64 is a graphical user interface showing two parameters encodedinto a leading edge of a signal, representing an embodiment of theinvention.

FIG. 65 are graphs of an encoded signal of FIG. 64 and a reflectedsignal of the encoded signal, representing an embodiment of theinvention.

FIG. 66 are natural log graphs of an exponential portion of a lowerportion of the leading edge on the encoded signal of FIG. 65,representing an embodiment of the invention.

FIG. 67 are graphs of the output of ratio of filters used on the encodedsignal of FIG. 65, representing an embodiment of the invention.

FIG. 63A are graphs of an encoded signal of FIG. 64 and a reflectedsignal of the encoded signal, representing an embodiment of theinvention.

FIG. 68B are natural log graphs of an exponential portion of an upperportion of the leading edge on the encoded signal of FIG. 63A,representing an embodiment of the invention.

FIG. 68C are graphs of the output of ratio of filters used on theencoded signal of FIG. 63A, representing an embodiment of the invention.

FIG. 69 is an applied signal and a reflected signal without bias,representing an embodiment of the invention.

FIG. 70 depicts the natural log of the signals in FIG.

69, representing an embodiment of the invention.

FIGS. 71A and 71B are graphs of a signal with a positive bias,representing an embodiment of the invention.

FIGS. 72A and 72B are graphs of a signal with a negative bias,representing an embodiment of the invention.

FIG. 73 is a graph of bias compensations for a plurality of signals,representing an embodiment of the invention.

FIGS. 74A and 74B are graphs of plurality of time of flight measurementsand bias values for those measurements, representing an embodiment ofthe invention.

FIG. 75 is a graph of measured constant threshold time domainreflectometer time of flight measurements, representing an embodiment ofthe invention.

FIG. 76 is a histogram of measured constant threshold time domainreflectometer time of flight measurements, representing an embodiment ofthe invention.

DETAILED DESCRIPTION

The invention and the various features and advantageous details thereofare explained more fully with reference to the nonlimiting embodimentsthat are illustrated in the accompanying drawings and detailed in thefollowing description. Descriptions of well known starting materials,processing techniques, components and equipment are omitted so as not tounnecessarily obscure the invention in detail. It should be understood,however, that the detailed description and the specific examples, whileindicating specific embodiments of the invention, are given by way ofillustration only and not by way of limitation. Various substitutions,modifications, additions and/or rearrangements within the spirit and/orscope of the underlying inventive concept will become apparent to thoseskilled in the art from this disclosure.

Within this application several publications are referenced by Arabicnumerals within parentheses or brackets. Full citations for these, andother, publications may be found at the end of the specificationimmediately preceding the claims after the section heading References.The disclosures of all these publications in their entireties are herebyexpressly incorporated by reference herein for the purpose of indicatingthe background of the invention and illustrating the state of the art.

In general, the context of the invention can include signaltransmission. The context of the invention can include a method and/orapparatus for measuring and estimating transmission line parameters. Thecontext of the invention can also include a method and/or apparatus forperforming high-speed communication over a lossy, dispersivetransmission line.

A practical application of the invention that has value within thetechnological arts is in time domain reflectometry. Further, theinvention is useful in conjunction with lossy transmission lines. Theinvention is also useful in conjunction with frequency dependenttransmission lines. Another practical application of the invention thathas value within the technological arts is high-speed communications.There are virtually innumerable other uses for the invention. In fact,any application involving the transmission of a signal may benefit.

A method for transmitting a waveform having a controllable attenuationand propagation velocity, representing an embodiment of the invention,can be cost effective and advantageous. The invention improves qualityand/or reduces costs compared to previous approaches.

The invention includes a method and/or apparatus for utilizing atruncated waveform, coined a Speedy Delivery (SD) waveform by theinventors. An analysis of SD propagation in coupled lossy transmissionlines is presented and practical considerations associated withtruncating the SD waveforms are addressed. Parameters used to describethe propagation of the SD waveform are defined and techniques fordetermining their values are presented. These parameters include theSpeedy Delivery propagation velocity, v_(SD), the Speedy Deliveryattenuation coefficient, A_(SD), and the Speedy Delivery impedance,Z_(SD). These parameters may depend on properties of a transmissionmedia as well as an exponential coefficient α (SD input signalparameter).

The invention describes SD signal propagation in coupled transmissionlines. An embodiment involving two transmission lines is illustrated.More complex configurations that include a larger number of coupledlines may be readily developed by one of ordinary skill in the art inlight of this disclosure. Additionally, the behavior of the propagationof a slowly varying envelope of an optical pulse containing a truncatedSD signal in a single mode communication fiber is described.

The invention teaches a method for using SD test pulses to obtainempirical estimates of the SD signal propagation parameters, A_(SD) andv_(SD), in lossy transmission lines, including those with frequencydependent parameters. Once the value of v_(SD) is determined for acalibration cable length, constant threshold time of flight measurementscan be used to measure distances to discontinuities in similar cables.The calibration procedure can be repeated experimentally for eachexponential coefficient α.

The invention provides a method for utilizing a wideband, non-SDwaveform in order to develop an empirical transfer function for acalibration length of a given transmission media. This transfer functionmay be used to simulate the transmission of SD waveforms with a widerange of values for the exponential coefficient α. The SD parametersA_(SD)(α) and v_(SD)(α) can be determined from these simulatedwaveforms. The inclusion of a simulation step in the calibration processcan significantly reduce the experimental measurements needed todetermine the values of A_(SD)(α) and v_(SD)(α) for a variety of valuesfor α. The SD behavior of a media can then be determined by simulation,without the need for an SD waveform generator.

The invention teaches a method for determining the SD impedance of atransmission line. It also illustrates the predicable variation ofZ_(SD) as a function of α. This predictability can allow a designer tocontrol the transmission line SD impedance by appropriately selectingthe parameter α.

The invention also describes an alternate precision distance measurementmethod using truncated SD test pulses with high loss, long lines. As thetest pulses travel down these lines, the attenuation can be so extensivethat their amplitudes are too small for common threshold crossing timeof flight measurements to be feasible. The invention includes a methodfor predicting an attenuation and overcoming this difficulty, extractingaccurate length measurements for long lossy lines.

The invention demonstrates the utility of the SD test waveform inaccurately determining distances to impedance discontinuities. Examplesshow how to use SD wave propagation as the basis for an accurate timedomain reflectometry (TDR) unit. This unit may include an SD waveformgenerator, a means of coupling with the transmission media, a means ofmeasuring the applied and reflected waveforms, a display, memorystorage, and computational ability to analyze and interpret the SDwaveforms. The invention can also include a computer. The invention caninclude a method which can be applied to modify currently available TDRunits to increase their accuracy. This modification may be a TDRsoftware modification resulting in a nonrecurring unit manufacturingcost. A standard TDR waveform may be used to determine an empiricaltransfer function for the line under test, and this transfer functionmay then be used to simulate the propagation of the SD waveform. Thisprocess allows all the advantages of SD to be incorporated intocurrently available units with only a modification of existingcomputational algorithms, creating a virtual SD TDR unit.

The truncated SD test pulse approach can also be applied to thepropagation of acoustic waves in sonar and geophysical applications.Reflected acoustic signals can be used to accurately determine thelocation of underwater objects and to determine locations of geophysicalstrata boundaries. Empirical transfer functions can be determined andused to simulate SD pulse propagation eliminating the need to physicallygenerate complex acoustic pulses.

The invention provides an SD waveform that has several propagationproperties in lossy, frequency dependent transmission media, which makeit suitable for enhancing a data transmission scheme.

The invention provides a method to increase the number of bitstransmitted per pulse by varying the value of α to encode data ontruncated SD waveforms. The example provided utilizes SD pulses withfour different α's sequentially transmitted on a 2 kft cable. These SDpulses are received at the end of the cable with the exponentialconstants undistorted. The four pulses may be transmitted as positive ornegative signals resulting in eight possible states. Therefore, eachpulse represents three bits of transmitted information in each symbolperiod. This strategy can be used with short, low loss transmissionlines.

The invention also teaches the use of SD propagation properties togenerate a set of pulses consisting of linear combinations of truncatedSD waveforms that are orthogonal at the receiver. Data can be encoded onthese pulses by varying their amplitude, and these amplitude-modulatedpulses may be transmitted simultaneously on a transmission line. Theindividual pulse amplitudes are computed at the receiver takingadvantage of their orthogonality. This method provides improved noiseimmunity, and can be used with longer, higher loss transmission lines.

The invention also teaches the use of SD waveforms in digital circuits,specifically, on-chip clock circuits. It provides a method for utilizingtruncated SD clock waveforms to reduce clock skew.

The propagation of a Speedy Delivery (SD) waveform in a transmissionline can be described by the equation V(x,t)=De⁻ ^(SD) ^([x−v) ^(SD)^(t]). The associated boundary condition at the input to the line isV(x=0,t)=De^(αt), wherein A_(SD) is the exponential coefficientdescribing the attenuation of the SD signal as a function of distancealong the line at a fixed time instant, and v_(SD) is the SD signalpropagation velocity. These quantities are also related by the simpleequation, A_(SD)(α)·v_(SD)(α)=α. The parameters A_(SD) and v_(SD) of thepropagating wave and the line boundary condition are all functions of α,and may be determined experimentally for a transmission line.

For the case of two balanced twisted wire pair transmission lines, thedifferential voltage can be equally divided between the two wires andthe current on each wire in the pair is equal in magnitude but oppositein direction. In this case, the telegrapher's equations can be written:

$\begin{matrix}{{\frac{\partial}{\partial x}V} = {Z \cdot I}} \\{{\frac{\partial}{\partial x}I} = {Y \cdot V}} \\{{\frac{\partial}{\partial x}\begin{bmatrix}{V_{1}\left( {x,t} \right)} \\{V_{2}\left( {x,t} \right)}\end{bmatrix}} = {\begin{bmatrix}{R_{1} + {L_{1}\frac{\partial}{\partial t}}} & {M_{12}\frac{\partial}{\partial t}} \\{M_{21}\frac{\partial}{\partial t}} & {R_{2} + {L_{2}\frac{\partial}{\partial t}}}\end{bmatrix} \cdot {\begin{bmatrix}{I_{1}\left( {x,t} \right)} \\{I_{2}\left( {x,t} \right)}\end{bmatrix}.}}} \\{{\frac{\partial}{\partial x}\begin{bmatrix}{I_{1}\left( {x,t} \right)} \\{I_{2}\left( {x,t} \right)}\end{bmatrix}} = {\begin{bmatrix}{G_{1} + {C_{1}\frac{\partial}{\partial t}}} & {C_{12}\frac{\partial}{\partial t}} \\{C_{21}\frac{\partial}{\partial t}} & {G_{2} + {C_{2}\frac{\partial}{\partial t}}}\end{bmatrix} \cdot \begin{bmatrix}{V_{1}\left( {x,t} \right)} \\{V_{2}\left( {x,t} \right)}\end{bmatrix}}}\end{matrix}$This analysis includes the net cross capacitance, C_(ij), and the netcross inductances, M_(ij). The cross resistance and conductance termsare neglected. These equations are then Laplace transformed into thecomplex frequency domain.

$\begin{matrix}{{\frac{\partial}{\partial x}\begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}} = {\begin{bmatrix}{{{\overset{\_}{R}}_{1}(s)} + {s{{\overset{\_}{L}}_{1}(s)}}} & {s{{\overset{\_}{M}}_{12}(s)}} \\{s{{\overset{\_}{M}}_{21}(s)}} & {{{\overset{\_}{R}}_{2}(s)} + {s{{\overset{\_}{L}}_{2}(s)}}}\end{bmatrix} \cdot \begin{bmatrix}{I_{1}\left( {x,s} \right)} \\{I_{2}\left( {x,s} \right)}\end{bmatrix}}} \\{{\frac{\partial}{\partial x}\begin{bmatrix}{I_{1}\left( {x,s} \right)} \\{I_{2}\left( {x,s} \right)}\end{bmatrix}} = {\begin{bmatrix}{{{\overset{\_}{G}}_{1}(s)} + {s{{\overset{\_}{C}}_{1}(s)}}} & {s{{\overset{\_}{C}}_{12}(s)}} \\{s{{\overset{\_}{C}}_{21}(s)}} & {{{\overset{\_}{G}}_{2}(s)} + {s{{\overset{\_}{C}}_{2}(s)}}}\end{bmatrix} \cdot \begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}}}\end{matrix}$Taking the partial derivative of the top equation with respect to x, andsubstituting the bottom equation for the

$\frac{\partial}{\partial x}\begin{bmatrix}{I_{1}\left( {x,s} \right)} \\{I_{2}\left( {x,s} \right)}\end{bmatrix}$term, gives the result:

${\frac{\partial^{2}}{\partial x^{2}}{V\left( {x,s} \right)}} = {{\overset{\_}{Z}(s)}{{\overset{\_}{Y}(s)} \cdot {V\left( {x,s} \right)}}}$${\frac{\partial^{2}}{\partial x^{2}}\begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}} = {\begin{bmatrix}{{{\overset{\_}{R}}_{1}(s)} + {s{{\overset{\_}{L}}_{1}(s)}}} & {s{{\overset{\_}{M}}_{12}(s)}} \\{s{{\overset{\_}{M}}_{21}(s)}} & {{{\overset{\_}{R}}_{2}(s)} + {s{{\overset{\_}{L}}_{2}(s)}}}\end{bmatrix} \cdot {\quad{{{\begin{bmatrix}{{{\overset{\_}{G}}_{1}(s)} + {s{{\overset{\_}{C}}_{1}(s)}}} & {s{{\overset{\_}{C}}_{12}(s)}} \\{s{{\overset{\_}{C}}_{21}(s)}} & {{{\overset{\_}{G}}_{2}(s)} + {s{{\overset{\_}{C}}_{2}(s)}}}\end{bmatrix} \cdot \begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}}{or}},{{\frac{\partial^{2}}{\partial x^{2}}{V\left( {x,s} \right)}} = {{{{\overset{\_}{\Gamma}(s)}^{2} \cdot {V\left( {x,s} \right)}}{\frac{\partial^{2}}{\partial x^{2}}\begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}}} = {\begin{bmatrix}{\left( {{{\overset{\_}{R}}_{1}(s)} + {s{{\overset{\_}{L}}_{1}(s)}}} \right)\left( {{{\overset{\_}{G}}_{1}(s)} + {s{{\overset{\_}{C}}_{1}(s)}}} \right)} & {s^{2}{{\overset{\_}{M}}_{12}(s)}{{\overset{\_}{C}}_{21}(s)}} \\{s^{2}{{\overset{\_}{M}}_{21}(s)}{{\overset{\_}{C}}_{12}(s)}} & {\left( {{{\overset{\_}{R}}_{2}(s)} + {s{{\overset{\_}{L}}_{2}(s)}}} \right)\left( {{{\overset{\_}{G}}_{2}(s)} + {s{{\overset{\_}{C}}_{2}(s)}}} \right)}\end{bmatrix} \cdot \begin{bmatrix}{V_{1}\left( {x,s} \right)} \\{V_{2}\left( {x,s} \right)}\end{bmatrix}}}}}}}$This system of equations has the solution:V(x,s)=e ^(− Γ(s)·x) ·A+e ^(+ Γ(s)·x) ·B

Applying the boundary conditions for a semi-infinite pair oftransmission lines, V(x=0,s) at x=0, and V(x→∞,s)=0 gives:V(x,s)=e ^(− Γ(s)·x) ·V(x=0,s).

Because Γ is a complex symmetric matrix, it can be written Γ=(ZY)^(1/2)= PγP ⁻¹, where P is the eigenvector matrix of the ZY productand γ is the diagonal eigenvalue matrix of Γ. Therefore, we can writee^(− Γ(s)·x)=P(s)·e^(− γ(s)·x)·P(s)⁻¹. This gives the result:V(x,s)=P(s)·e ^(− γ(s)·x) ·P(s)⁻¹ ·V(x=0,s)

For the case where the applied boundary conditions at x=0 are SDwaveforms, V₁(x=0,t)=D₁e^(α) ¹ ^(t), V₂(x=0,t)=D₂e^(α2) ^(t), the SDsolution is:

${{V\left( {x,t} \right)} = {{{P\left( \alpha_{1} \right)} \cdot {\mathbb{e}}^{{- {\gamma{(\alpha_{1})}}} \cdot x} \cdot {P^{- 1}\left( \alpha_{1} \right)} \cdot \begin{bmatrix}{D_{1} \cdot {\mathbb{e}}^{\alpha_{1} \cdot t}} \\0\end{bmatrix}} + {{P\left( \alpha_{2} \right)} \cdot {\mathbb{e}}^{{- {\gamma{(\alpha_{2})}}} \cdot x} \cdot {P^{- 1}\left( \alpha_{2} \right)} \cdot \begin{bmatrix}0 \\{D_{2} \cdot {\mathbb{e}}^{\alpha_{2} \cdot t}}\end{bmatrix}}}},$where P(α_(i)) is a real matrix.

${V\left( {x,t} \right)} = {{\begin{bmatrix}{P_{11}\left( \alpha_{1} \right)} & {P_{12}\left( \alpha_{1} \right)} \\{P_{21}\left( \alpha_{1} \right)} & {P_{22}\left( \alpha_{1} \right)}\end{bmatrix} \cdot \begin{bmatrix}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{1})}}} \cdot x} & 0 \\0 & {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{1})}}} \cdot x}\end{bmatrix} \cdot {\begin{bmatrix}{P_{11}\left( \alpha_{1} \right)} & {P_{12}\left( \alpha_{1} \right)} \\{P_{21}\left( \alpha_{1} \right)} & {P_{22}\left( \alpha_{1} \right)}\end{bmatrix}^{- 1}.\begin{bmatrix}{D_{1} \cdot {\mathbb{e}}^{\alpha_{1} \cdot t}} \\0\end{bmatrix}}} + {\begin{bmatrix}{P_{11}\left( \alpha_{2} \right)} & {P_{12}\left( \alpha_{2} \right)} \\{P_{21}\left( \alpha_{2} \right)} & {P_{22}\left( \alpha_{2} \right)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{2})}}} \cdot x} & 0 \\0 & {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{2})}}} \cdot x}\end{bmatrix} \cdot \begin{bmatrix}{P_{11}\left( \alpha_{2} \right)} & {P_{12}\left( \alpha_{2} \right)} \\{P_{21}\left( \alpha_{2} \right)} & {P_{22}\left( \alpha_{2} \right)}\end{bmatrix}^{- 1} \cdot \begin{bmatrix}0 \\{D_{2} \cdot {\mathbb{e}}^{\alpha_{2} \cdot t}}\end{bmatrix}}}}}$If we designate P⁻¹ as Q, we can write:

$\left. {{{V\left( \quad \right.}x}, t} \right) = {\begin{bmatrix}{P_{11}\left( \alpha_{1} \right)} & {P_{12}\left( \alpha_{1} \right)} \\{P_{21}\left( \alpha_{1} \right)} & {P_{22}\left( \alpha_{1} \right)}\end{bmatrix} \cdot \begin{bmatrix}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{1})}}} \cdot x} & 0 \\0 & {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{1})}}} \cdot x}\end{bmatrix} \cdot {\quad{{\begin{bmatrix}{Q_{11}\left( \alpha_{1} \right)} & {Q_{12}\left( \alpha_{1} \right)} \\{Q_{21}\left( \alpha_{1} \right)} & {Q_{22}\left( \alpha_{1} \right)}\end{bmatrix} \cdot \begin{bmatrix}{D_{1} \cdot {\mathbb{e}}^{\alpha_{1} \cdot t}} \\0\end{bmatrix}} + {\begin{bmatrix}{P_{11}\left( \alpha_{2} \right)} & {P_{12}\left( \alpha_{2} \right)} \\{P_{21}\left( \alpha_{2} \right)} & {P_{22}\left( \alpha_{2} \right)}\end{bmatrix} \cdot {\quad{\begin{bmatrix}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{2})}}} \cdot x} & 0 \\0 & {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{2})}}} \cdot x}\end{bmatrix} \cdot {\quad{{{\begin{bmatrix}{Q_{11}\left( \alpha_{2} \right)} & {Q_{12}\left( \alpha_{2} \right)} \\{Q_{21}\left( \alpha_{2} \right)} & {Q_{22}\left( \alpha_{2} \right)}\end{bmatrix} \cdot \begin{bmatrix}0 \\{D_{2} \cdot {\mathbb{e}}^{\alpha_{2} \cdot t}}\end{bmatrix}}{or}},{{V\left( {x,t} \right)} = {{{P\left( \alpha_{1} \right)} \cdot \begin{bmatrix}{{Q_{11}\left( \alpha_{1} \right)} \cdot {\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{1})}}} \cdot x}} \\{{Q_{21}\left( \alpha_{1} \right)} \cdot {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{1})}}} \cdot x}}\end{bmatrix} \cdot D_{1} \cdot {\mathbb{e}}^{\alpha_{1} \cdot t}} + {{{P\left( \alpha_{2} \right)} \cdot \begin{bmatrix}{{Q_{12}\left( \alpha_{2} \right)} \cdot {\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{2})}}} \cdot x}} \\{{Q_{22}\left( \alpha_{2} \right)} \cdot {\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{2})}}} \cdot x}}\end{bmatrix} \cdot D_{2} \cdot {\mathbb{e}}^{\alpha_{2} \cdot t}}{or}}}},{{V\left( {x,t} \right)} = {{\begin{bmatrix}{{{Q_{11}\left( \alpha_{1} \right)}{P_{11}\left( \alpha_{1} \right)}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{1})}}} \cdot x}} + {{Q_{21}\left( \alpha_{1} \right)}{P_{12}\left( \alpha_{1} \right)}{\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{1})}}} \cdot x}}} \\{{{Q_{11}\left( \alpha_{1} \right)}{P_{21}\left( \alpha_{1} \right)}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{1})}}} \cdot x}} + {{Q_{21}\left( \alpha_{1} \right)}{P_{22}\left( \alpha_{1} \right)}{\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{1})}}} \cdot x}}}\end{bmatrix} \cdot D_{1} \cdot {\mathbb{e}}^{\alpha_{1} \cdot t}} + {\quad{\begin{bmatrix}{{{Q_{12}\left( \alpha_{2} \right)}{P_{11}\left( \alpha_{2} \right)}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{2})}}} \cdot x}} + {{Q_{22}\left( \alpha_{2} \right)}{P_{12}\left( \alpha_{2} \right)}{\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{2})}}} \cdot x}}} \\{{{Q_{12}\left( \alpha_{2} \right)}{P_{21}\left( \alpha_{2} \right)}{\mathbb{e}}^{{- {\gamma_{1}{(\alpha_{2})}}} \cdot x}} + {{Q_{22}\left( \alpha_{2} \right)}{P_{22}\left( \alpha_{2} \right)}{\mathbb{e}}^{{- {\gamma_{2}{(\alpha_{2})}}} \cdot x}}}\end{bmatrix} \cdot D_{2} \cdot {{\mathbb{e}}^{\alpha_{2} \cdot t}.}}}}}}}}}}}}}$

The real constants associated with the eigenvector matrix, P, itsinverse, P⁻¹=Q, and the SD parameters D can be written:A ₁₁ =D ₁ P ₁₁(α₁)Q ₁₁(α₁)A ₁₂ =D ₁ P ₁₂(α₁)Q ₂₁(α₁)A ₂₁ =D ₂ P ₁₁(α₂)Q ₁₂(α₂)A ₂₂ =D ₂ P ₁₂(α₂)Q ₂₂(α₂)B ₁₁ =D ₁ P ₂₁(α₁)Q ₁₁(α₁)B ₁₂ =D ₁ P ₂₂(α₁)Q ₂₁(α₁)B ₂₁ =D ₂ P ₂₁(α₂)Q ₁₂(α₂)B ₂₂ =D ₂ P ₂₂(α₂)Q ₂₂(α₂)The signals on transmission lines will be:V ₁(x,t)=A ₁₁ ·e ^(−γ) ¹ ^((α) ¹ ^()·x) e ^(α) ¹ ^(·t) +A ₂₁ ·e ^(γ) ²^((α) ¹ ^()·x) e ^(α) ¹ ^(·t) +A ₁₂ ·e ^(−γ) ¹ ^((α) ² ^()·x) e ^(α) ²^(·t) +A ₂₂ ·e ^(−γ) ² ^((α) ² ^()·x) e ^(α) ² ^(·t)V ₂(x,t)=B ₁₁ ·e ^(−γ) ¹ ^((α) ¹ ^()·x) e ^(α) ¹ ^(·t) +B ₂₁ ·e ^(−γ) ²^((α) ¹ ^()·x) e ^(α) ¹ ^(·t) +B ₁₂ ·e ^(−γ) ¹ ^((α) ² ^()·x) e ^(α) ²^(·t) +B ₂₂ ·e ^(−γ) ² ^((α) ² ^()·x) e ^(α) ² ^(·t)or,

V₁(x, t) = A₁₁ ⋅ 𝕖^(−A_(SD₁₁)(x − v_(SD₁₁)t)) + A₂₁ ⋅ 𝕖^(−A_(SD₂₁)(x − v_(SD₂₁)t)) + A₁₂ ⋅ 𝕖^(−A_(SD₁₂)(x − v_(SD 12)t)) + A₂₂ ⋅ 𝕖^(−A_(SD₂₂)(x − v_(SD₂₂)t))V₂(x, t) = B₁₁ ⋅ 𝕖^(−A_(SD₁₁)(x − v_(SD₁₁)t)) + B₂₁ ⋅ 𝕖^(−A_(SD₂₁)(x − v_(SD₂₁)t)) + B₁₂ ⋅ 𝕖^(−A_(SD₁₂)(x − v_(SD₁₂)t)) + B₂₂ ⋅ 𝕖^(−A_(SD₂₂)(x − v_(SD₂₂)t))

Where:

A_(SD) ₁₁ =γ₁(α₁), A_(SD) ₁₂ =γ₁(α₂), A_(SD) ₂₁ =γ₂(α₁), and A_(SD) ₂₂=γ₂(α₂).

$\begin{matrix}{{v_{{SD}_{11}} = \frac{\alpha_{1}}{\gamma_{1}\left( \alpha_{1} \right)}},{v_{{SD}_{12}} = \frac{\alpha_{2}}{\gamma_{1}\left( \alpha_{2} \right)}},} \\{{v_{{SD}_{21}} = \frac{\alpha_{1}}{\gamma_{2}\left( \alpha_{1} \right)}},{{{and}\mspace{14mu} v_{{SD}_{22}}} = {\frac{\alpha_{2}}{\gamma_{2}\left( \alpha_{2} \right)}.}}}\end{matrix}$

Thus, there are SD attenuations and velocities associated with eacheigenvalue, γ_(i), of the product ZY evaluated at each α.

Because positive exponential waveforms continually increase, practicalconsiderations may create the need to truncate or limit the waveform atsome level determined by the specific application. Truncation may beaccomplished by several methods, as is known in the art.

The result of this truncation of the input of a transmission media isthat the propagating exponential (SD) signal is also limited inmagnitude as it travels in the media, and in lossy media, this maximumamplitude decreases with distance traveled exhibiting an attenuatingbehavior.

The propagating (SD) signal is described in the coordinate frame, (x,t), by V(x,t)=De^(−A) ^(SD) ^((α)[x−v) ^(SD) ^((α)·t])=De^(α·t)e^(−A)^(SD) ^((α)·x), where A_(SD)(α)v_(SD)(α)=α.

Referring to FIG. 1A, a graph of an SD waveform for (x=0,t) is depicted.The truncated input waveform at x=0 reaching a limit at t=t_(i) isshown.

In the case of a four parameter (LCRG) transmission line, we have theexpressions

$\begin{matrix}{{{A_{SD}(\alpha)} = \sqrt{{{LC}\;\alpha^{2}} + {\left( {{RC} + {LG}} \right)\alpha} + {RG}}},{and}} \\{{v_{SD}(\alpha)} = {\frac{\alpha}{\sqrt{{{LC}\;\alpha^{2}} + {\left( {{RC} + {LG}} \right)\alpha} + {RG}}}.}}\end{matrix}$

The attenuation of the traveling SD wave may be viewed from theperspective of a moving reference frame traveling at the speed1/√{square root over (LC)}.

Introducing the new coordinates:t′=t−x√{square root over (LC)}.x′=xThenV(x′,t′)=De ^(−A) ^(SD) ^((α)·[x′−v) ^(SD)^((α)·(t′+x′·√{square root over (LC)})]),orV(x′,t′)=De ^(α·t′) e ^(−A) ^(SD) ^((α)·[1−v) ^(SD)^((α)·√{square root over (LC)}]·x′).

If we view the traveling wave after traveling a distance, x=x′=l, in thetransmission media, thenV(l,t′)=De ^(α·t′) e ^(−A) ^(SD) ^((α)·[1−v) ^(SD)^((α)·√{square root over (LC)}]·l).

Setting t′=t_(i), yields the maximum value of the truncated SD waveformat x=l, and for any value 0≦t_(j)≦t_(i), the value of the signal at theassociated point t_(j) in the moving reference frame at x=l is:V(l,t _(j))=De ^(α·t) ^(j) e ^(−A) ^(SD) ^((α)·[1−v) ^(SD)^((α)·√{square root over (LC)}]·l)

Referring to FIG. 1B, a graph of an SD waveform in moving frame at x′=l(x=l,t′) is depicted. The attenuation of this truncated waveform isdefined by the ratio:

$\begin{matrix}{\frac{V\left( {l,t_{j}} \right)}{V\left( {0,t_{j}} \right)} = \frac{D\;{\mathbb{e}}^{\alpha \cdot t_{j}}{\mathbb{e}}^{{- {A_{SD}{(\alpha)}}} \cdot {\lfloor{1 - {{v_{SD}{(\alpha)}} \cdot \sqrt{LC}}}\rfloor} \cdot l}}{D\;{\mathbb{e}}^{\alpha \cdot t_{j}}}} \\{= {\mathbb{e}}^{{- {A_{SD}{(\alpha)}}} \cdot {\lbrack{1 - {{v_{SD}{(\alpha)}} \cdot \sqrt{LC}}}\rbrack} \cdot l}}\end{matrix}$

This attenuation of the signal at a point t_(j) of a truncated SDwaveform traveling in a transmission line with frequency dependentparameters whose Laplace transforms are L(s), C(s), R(s), and G(s), is:

$\frac{V\left( {l,t_{j}} \right)}{V\left( {0,t_{j}} \right)} = {\mathbb{e}}^{{- {{\overset{\_}{A}}_{SD}{(\alpha)}}} \cdot {\lfloor{1 - {{{\overset{\_}{v}}_{SD}{(\alpha)}} \cdot \sqrt{\overset{\_}{L}{\overset{\_}{C}{(\alpha)}}}}}\rfloor} \cdot l}$

where:

${{{\overset{\_}{A}}_{SD}(\alpha)} = \sqrt{{{\overset{\_}{LC}(\alpha)} \cdot \alpha^{2}} + {\left( {{\overset{\_}{RC}(\alpha)} + {\overset{\_}{LG}(\alpha)}} \right) \cdot \alpha} + {\overset{\_}{RG}(\alpha)}}},{and}$${{\overset{\_}{v}}_{SD}(\alpha)} = {\frac{\alpha}{\sqrt{{{\overset{\_}{LC}(\alpha)} \cdot \alpha^{2}} + {\left( {{\overset{\_}{RC}(\alpha)} + {\overset{\_}{LG}(\alpha)}} \right) \cdot \alpha} + {\overset{\_}{RG}(\alpha)}}}.}$

Furthermore, in this transmission media with frequency dependentparameters, if the input exponential pulse is rapidly closed when theexponential amplitude truncation limit is reached, then the end of theexponential region of the leading edge of the transmitted pulse isfurther reduced in amplitude and rounded by the chromatic dispersion.

Coupled transmission lines can be treated in a similar fashion. Theequations governing SD propagation on one of two coupled transmissionlines is:V₁(x,t)=A₁₁·e^(−A) _(SD)11 (X-v_(SD)Ilt)+A₂₁.e-A_(SD21) (x-v_(SD21)t)+Al₂-e A_(SD)I2 (X-v_(SD)2t)+A₂₂.e-A_(SD)2₂(X-v_(SD) t).

Applying the substitution: t′=t−x√{square root over ( LC, x′=x, givesthe result:

$\frac{V_{1}\left( {{x^{\prime} = l},{t^{\prime} = t_{i}}} \right)}{V_{1}\left( {{x = 0},{t = t_{i}}} \right)} = {\frac{\begin{matrix}{{A_{11} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}} \cdot {\mathbb{e}}^{{- {A_{{SD}_{11}}({1 - {v_{{SD}_{11}}\sqrt{\overset{\_}{LC}}}})}}l}} +} \\{A_{21} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}} \cdot {\mathbb{e}}^{{- {A_{{SD}_{21}}({1 - {v_{{SD}_{21}}\sqrt{\overset{\_}{LC}}}})}}l}}\end{matrix}}{\begin{matrix}{{A_{11} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}}} + {A_{21} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}}} +} \\{{A_{12} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}}} + {A_{22} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}}}}\end{matrix}} + \frac{\begin{matrix}{A_{12} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}} \cdot {\mathbb{e}}^{{{- {A_{{SD}_{12}}({1 - {v_{{SD}_{12}}\sqrt{\overset{\_}{LC}}}})}}l} +}} \\{A_{22} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}} \cdot {\mathbb{e}}^{{- {A_{{SD}_{22}}({1 - {v_{{SD}_{22}}\sqrt{\overset{\_}{LC}}}})}}l}}\end{matrix}}{\begin{matrix}{{A_{11} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}}} + {A_{21} \cdot {\mathbb{e}}^{\alpha_{1}t_{i}}} +} \\{{A_{12} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}}} + {A_{22} \cdot {\mathbb{e}}^{\alpha_{2}t_{i}}}}\end{matrix}}}$Which can be written:

$\frac{V_{1}\left( {l,t_{j}} \right)}{V_{1}\left( {0\; t_{j}} \right)} = {\frac{\begin{pmatrix}{A_{11} \cdot {\mathbb{e}}^{{{- {A_{{SD}_{11}}({1 - {v_{{SD}_{11}}\sqrt{\overset{\_}{LC}}}})}}l} +}} \\{A_{21} \cdot {\mathbb{e}}^{{- {A_{{SD}_{21}}({1 - {v_{{SD}_{21}}\sqrt{\overset{\_}{LC}}}})}}l}}\end{pmatrix}}{\begin{matrix}{\left( {A_{11} + A_{21}} \right) + {\left( {A_{12} + A_{22}} \right) \cdot}} \\{\mathbb{e}}^{{({\alpha_{2} - \alpha_{1}})}t_{i}}\end{matrix}} + \frac{\begin{pmatrix}{{A_{12} \cdot {\mathbb{e}}^{{- {A_{{SD}_{12}}({1 - {v_{{SD}_{12}}\sqrt{\overset{\_}{LC}}}})}}l}} +} \\{A_{22} \cdot {\mathbb{e}}^{{- {A_{{SD}_{22}}({1 - {v_{{SD}_{22}}\sqrt{\overset{\_}{LC}}}})}}l}}\end{pmatrix}}{\begin{matrix}{{\left( {A_{11} + A_{21}} \right) \cdot {\mathbb{e}}^{{({\alpha_{1} - \alpha_{2}})}t_{i}}} +} \\\left( {A_{12} + A_{22}} \right)\end{matrix}}}$

Another example includes of the evolution of a slowly varying envelope,E(x,t), of an optical pulse in a single mode communication fiber. TheSchrödinger partial differential equation [3],

${{\frac{\partial E}{\partial x} + {\beta_{1}\frac{\partial E}{\partial t}} - {\frac{j}{2}\beta_{2}\frac{\partial^{2}E}{\partial t^{2}}} - {\frac{1}{6}\beta_{3}\frac{\partial^{3}E}{\partial t^{3}}}} = {{- \frac{\gamma}{2}}E}},$describes the evolution of the shape of the propagating pulse envelopeundergoing chromatic dispersion in fiber. Chromatic dispersion occursbecause the mode-propagation constant, β(ω), is a nonlinear function ofthe angular frequency ω, where

$\beta_{n} = \left\lbrack \frac{\partial^{n}\beta}{\partial\omega^{n}} \right\rbrack_{\omega = \omega_{o}}$

and ω_(o) is the frequency of the light being modulated in the fiber.

ω_(o)/β_(o) is the phase velocity of the pulse. 1/β₁ is the groupvelocity, β₂ is the group velocity dispersion (GVD) parameter whichcauses the pulse to broaden as it propagates in the fiber. β₃ is thethird-order dispersion (TOD) parameter [4]

The higher order derivatives of β with respect to ω are assumednegligible. However, they may be added in these analyses by anyonefamiliar with the art. The fiber attenuation is represented by theparameter γ.

The SD solution of the linear Schrödinger equation with loss (γ≠0) is:

${E\left( {x,t} \right)} = {D\;{\mathbb{e}}^{\lbrack{\frac{t}{T_{o}} - {{({\frac{\beta_{1}}{T_{o}} - \frac{\beta_{3}}{6\; T_{o}^{3}} + \frac{\gamma}{2}})} \cdot x}}\rbrack}{{\mathbb{e}}^{j(\frac{\beta_{2} \cdot x}{2\; T_{o}^{2}})}.}}$

The boundary condition at the input to the fiber at x 0 is

${{E\left( {0,t} \right)} = {D\;{\mathbb{e}}^{\frac{t}{To}}}},$(see FIG. 1A with

$\left. {\alpha = \frac{1}{T_{o}}} \right).$The receiver detector in a fiber optic communication network responds tothe square of the magnitude |E(x,t)|² of the propagating pulse envelope.In this case|E(x,t)|² =D ² e ^(−2·A(T) ^(o) ^()[x−v) ^(sd) ^((T) ^(o) ^()·t]),

where the velocity of propagation of a truncated SD leading edge of theenvelope is

${{{v_{sd}\left( T_{o} \right)} = \frac{1}{\left( {\beta_{1} - \frac{\beta_{3}}{6\; T_{o}^{2}} + \frac{T_{o}\gamma}{2}} \right)}},{{{and}\mspace{14mu}{A\left( T_{o} \right)}} = {{{\left( {\frac{\beta_{1}}{T_{o}} - \frac{\beta_{3}}{6\; T_{o}^{3}} + \frac{\gamma}{2}} \right).\;{Note}}\mspace{14mu}{that}\mspace{14mu}{{v_{sd}\left( T_{o} \right)} \cdot {A\left( T_{o} \right)}}} = {\frac{1}{T_{o}}.}}}}\mspace{11mu}$

Transforming to a moving reference frame traveling at the group velocity1/β₁,t′=t−β ₁ xx′=x

The SD propagation speed in this moving reference frame is

${{{{\overset{\_}{v}}_{sd}\left( T_{o} \right)}❘_{{i\; n\mspace{14mu} x^{\prime}},{t^{\prime}\mspace{14mu}{frame}}}} = \frac{1}{T_{o}\left( {\frac{\gamma}{2} - \frac{\beta_{3}}{6\; T_{o}^{3}}} \right)}},$

The attenuation of the truncated SD portion of the leading edge of thepulse envelope is

$\frac{{{E\left( {l,t_{j\;}} \right)}}^{2}}{{{E\left( {0,t_{j}} \right)}}^{2}} = {{\mathbb{e}}^{{- {({\frac{\gamma}{2} - \frac{\beta_{3}}{6\; T_{o}^{3}}})}}l}.}$

This result implies that if T_(o) can be made small enough, thisattenuation may be reduced. Higher order terms of β(ω), which were notincluded is this particular analysis, may become more significant asT_(o) become smaller.

EXAMPLES

Specific embodiments of the invention will now be further described bythe following, nonlimiting examples which will serve to illustrate insome detail various features. The following examples are included tofacilitate an understanding of ways in which the invention may bepracticed. It should be appreciated that the examples which followrepresent embodiments discovered to function well in the practice of theinvention, and thus can be considered to constitute preferred modes forthe practice of the invention. However, it should be appreciated thatmany changes can be made in the exemplary embodiments which aredisclosed while still obtaining like or similar result without departingfrom the spirit and scope of the invention. Accordingly, the examplesshould not be construed as limiting the scope of the invention.

Example 1 Method for Determining SD Waveform Parameters, A_(SD) andv_(SD)

This example teaches how SD test pulses may be used to obtain empiricalestimates of the transmission line parameters, A_(SD) and v_(SD), thatdescribe the propagation of SD waveforms in lossy, dispersive lines,including those with frequency dependent parameters. The numericalvalues of A_(SD) and v_(SD) as a function of α can be determinedempirically.

Referring to FIG. 2, an experimental set up that may be used todetermine these parameters for a copper, twisted wire pair transmissionline is depicted. The test length of the transmission line, d, and thevalue of α for the applied SD input signal are measured. An additionallength of transmission line that is identical to the line under test isattached as shown and terminated with an open circuit. In this examplethe cables are two 24 AWG, individually shielded, twisted wire pairs ina 1002 ft coil of T1 cable.

The values of D, α and the duration of the SD signal were chosen toprevent the occurrence of reflections at the measurement point, d,during the initial time of flight of the SD signal propagating in thetest line. The additional length of transmission line also ensures thatall reflections occurring in the test line are delayed until well afterthe propagating SD waveform measurement at d is complete.

The SD signal time of flight can be directly measured by timing thepropagating SD waveform crossing of constant voltage thresholds (FIG.3). The value of the SD signal propagation velocity in the line, v_(SD),can be calculated using the relationship, TOF=(t_(f)−t_(o))=v_(sd)d. Thetimes of flight for these waveforms shown again in FIG. 4A at thresholdsfrom 1.2 volts to 4.8 volts are plotted in FIG. 4B. The average TOF forthe threshold overlap region shown in FIG. 4.3 b is 1,716 ns. This TOFresults in

$v_{SD} = {\frac{d}{{TOF}_{AVG}} = {\frac{1002\mspace{14mu}{ft}}{1,716 \times 10^{- 9}\mspace{14mu}\sec} = {5.839 \times 10^{8}{\frac{ft}{\sec}.}}}}$

The SD attenuation coefficient, A_(SD), can be calculated from α andv_(sd) using the relationship A_(SD)·v_(SD)=α. The calculated A_(SD)obtained from the waveforms shown in FIG. 4.3 a is

$A_{SD} = {\frac{\alpha}{v_{SD}} = {\frac{3.0 \times 10^{6}\mspace{11mu}{1/\sec}}{5.839 \times 10^{8}\mspace{14mu}{{ft}/\sec}} = {5.138 \times 10^{- 3}\mspace{14mu}{1/{{ft}.}}}}}$

Once this velocity is known for a calibrated length of the cable, thistype of threshold crossing TOF measurement can be used to determine theunknown length of another sample of the same transmission line.

Referring to FIG. 5A, the measured input voltage of the two twisted wirepairs in a T1 line connected in series is depicted (see FIG. 2) andone-half of the voltage measured at the end of the open terminatedsecond twisted wire pair in the T1 line (R_(t)=∞ in FIG. 2). The averageTOF for the threshold overlap region shown in FIG. 5B is 3,418 ns. Thistime of flight together with the previously calibrated v_(SD), resultsin an estimated distance of 1,996 ft, or 99.6% of the actual length(2004 ft).

FIG. 5A also indicates a potential difficulty in longer lines caused bythe attenuation of the truncated SD pulse. The region of SD thresholdoverlap grows smaller as the pulse propagates, and is attenuated overlonger distances. The accuracy of this measurement can be improved, andthe difficulty caused by the reduced region of threshold overlap can beovercome for very long transmission lines using the TOF measurementtechnique discussed in example 4.

The high accuracy of this method results from the use of an average ofTOF measurements taken over a range of signal threshold amplitudes. Itis feasible to use the average of these values to improve the accuracyof the measurement of TOF since all points on the leading SD edge of thewaveform travel at the same speed. This is in contrast with the currenttime domain reflectometry art in which a measurement of the time ofemergence of the dispersing test pulse is attempted. This is equivalentto attempting to accurately measure the threshold crossing time of thepulse at a zero threshold level when the slope of the pulse is alsonearly zero.

Example 2 Extracting SD Parameters from Empirical Transfer Function

The method described in example 1 can be limiting in that it should beexperimentally repeated for each exponential coefficient α. This mayrequire a lengthy laboratory calibration time for each cable type.Simulation can be included in the calibration process to significantlyreduce the experimental measurements needed to determine the values ofA_(SD) and v_(SD). The use of simulation in this process requires atransfer function that describes the response of the specific type ofcable. The α's analyzed for the T1 cable range from 1×10⁵ sec⁻¹ to 1×10⁷sec⁻¹. This range of a's corresponds to the frequency band of 16 kHz to1.6 MHz, so the transfer function must be accurate for this range. Thetransfer function of the cable may available from the manufacturer orcan be determined empirically by applying a known pulse and measuringthe applied cable input waveform and the response waveform at a known(calibration) distance along the cable. The transfer function is thencalculated from the ratio of the fast Fourier transforms (FFT) of theinput-output pulses:

${H_{est}\left( {j\;\omega} \right)} = {\frac{F\; F\;{T\left( V_{{Measured}\mspace{14mu}{at}\mspace{14mu} l} \right)}}{F\; F\;{T\left( V_{Applied} \right)}}.}$Once the transfer function is determined, the propagation of SDwaveforms with a series of different values for α can be simulated andthe times of flight and SD parameters, A_(SD) and v_(SD), determined.

This method can be demonstrated for the T1 cable used in example 1. Theend of the total 2004 length has been terminated with a 100-Ohm resistor(R_(t) in FIG. 4.1) to minimize reflections at the 1002 ft measurementpoint. FIG. 6A shows a measured input pulse applied to the T1. FIG. 6Bshows the output pulse measured at 1002 ft. FIGS. 7A-B show the PowerSpectral Density of the measured pulses. The range of α's is indicatedto show that there is sufficient energy in the frequency band ofinterest to empirically estimate an accurate transfer function. Themagnitude and the phase of the transfer function determined from themeasured signals are shown in FIG. 8.

Once the transfer function is known, a waveform with any α in theindicated range of interest can be simulated and the value of A_(SD) andv_(SD) determined. The values of A_(SD) and v_(SD) of the measured SDwaveforms of section three provide a test of the transfer function foran α of 3×10⁶ sec⁻¹. FIG. 9 shows the time of flight plots determinedusing the input signal measured in section two, and the 1002 ft waveformpredicted by simulation using the empirical transfer function. Thesimulation TOF result (FIG. 9) of 1,714 nsec is almost identical to theexperimentally measured (FIG. 5B) TOF of 1,716 nsec. FIG. 10 shows thedifference between the measured waveform at 1002 ft and the simulatedwaveform at 1002 ft (FIG. 9, Top). During the SD propagation region thedifference is on the order of +20 mV.

This process is performed for the range of a's by simulating the initialSD waveforms and using the transfer function to predict the waveforms at1002 ft. In all cases, the simulated initial waveforms were closed witha 1 μsec ramp to reduce their high frequency components. FIGS. 11-12show the simulated input and 1002 ft curves for the endpoint α's of1×10⁵ sec⁻¹ and 1×10⁷ sec⁻¹ along with their respective constantthreshold times of flight curves. The values of v_(SD) and A_(SD) fromsimulation are presented in FIGS. 13A-B, illustrated in Table I for therange of a's, as set forth below (* values in parenthesis obtained usingmethods of example 1, with measured input and output SD signals alongactual T1 Line):

TABLE I V_(SD) and A_(SD) Determined By Simulation with EmpiricalTransfer Function Alpha 1002 Simulation Simulation V_(SD) SimulationA_(SD) (sec⁻¹) TOF (μsec) (×10⁸ ft/sec) (×10⁻³ l/ft) 1 × 10⁵ 4.487 2.2330.448 2 × 10⁵ 3.068 3.266 0.612 3 × 10⁵ 2.608 3.842 0.781 4 × 10⁵ 2.3824.207 0.951 5 × 10⁵ 2.271 4.412 1.133 6 × 10⁵ 2.172 4.613 1.301 7 × 10⁵2.103 4.765 1.469 8 × 10⁵ 2.040 4.912 1.629 9 × 10⁵ 1.999 5.013 1.796 1× 10⁶ 1.973 5.079 1.969 2 × 10⁶ 1.792 5.592 3.577 3 × 10⁶ 1.714 (1.716*)5.849 (5.839*) 5.129 (5.138*) 4 × 10⁶ 1.666 6.014 6.651 5 × 10⁶ 1.6346.132 8.154 6 × 10⁶ 1.610 6.224 9.641 7 × 10⁶ 1.592 6.294 11.122 8 × 10⁶1.577 6.354 12.591 9 × 10⁶ 1.565 6.403 14.057 1 × 10⁷ 1.555 6.444 15.519

Example 3 Measurement of SD Transmission Line Impedance

The SD line impedance, Z_(SD)(α), is a real number which may bedetermined for the α's of interest. If the Laplace transform of thefrequency dependent transmission line parameters R(s), L(s), C(s), andG(s), are known, then Z_(SD)(α) may be computed from,

$Z_{SD} = {\sqrt{\frac{{\alpha{\overset{\_}{L}(\alpha)}} + {\overset{\_}{R}(\alpha)}}{{\alpha{\overset{\_}{C}(\alpha)}} + {\overset{\_}{G}(\alpha)}}}.}$

The SD line impedance, Z_(SD)(α), may also be experimentally determinedfor a test transmission line by measuring the SD portion of the voltagewaveform across various known termination impedances and computing thevalue Z_(SD)(α) from the measurements. The measured SD waveform at atermination, v_(SD)(R_(t)), consists of the sum of the incident SDwaveform, V_(SD) ⁺, and a reflected SD wave. The reflected wave is theproduct of the SD reflection coefficient, Γ, and the incident wave. Themeasured SD signal at the termination will be

${V_{SD}\left( R_{t} \right)} = {{\left( {1 + \Gamma_{t}} \right) \cdot V_{SD}^{+}} = {\frac{2 \cdot R_{t}}{R_{t} + Z_{SD}} \cdot {V_{SD}^{+}.}}}$Note that all lumped and distributed impedance values are real for SDsignals.

The experimental setup used to determine Z_(SD) from line measurementsis shown in FIG. 14. The measured input and termination waveforms of thevoltages for a 1002 ft of T1 line, are shown in FIG. 15A. Thesemeasurements were made with an α of 1.5×10⁶ l/sec and terminationresistances of 49.5 ohms, 98.6 ohms and an open circuit. The magnitudeof the waveform with no reflection, (the incident waveform V_(SD) ⁺), isdetermined by dividing the signal measured with an open termination,(+R_(t)=∞, Γ_(∞)=1), in half. The ratio of the waveforms

V_(SD)(R_(t))/V_(SD)⁺,with known terminations and the incident waveform gives a directmeasurement of the reflection coefficient, Γ_(t), for each terminationresistance,

$\Gamma_{t} = {\frac{V_{SD}\left( R_{t} \right)}{V_{SD}^{+}} - 1.}$FIG. 15B shows the measured ratios for this cable and α. Using therelationship for the transmission coefficient,

${{1 + \Gamma_{t}} = \frac{2 \cdot R_{t}}{R_{t} + Z_{SD}}},{{{gives}{\mspace{11mu}\;}Z_{SD}} = {\frac{R_{t}}{1 + \Gamma_{t}}{\left( {1 - \Gamma_{t}} \right).}}}$

Using the two finite values for R_(t), 98.6Ω and 49.5Ω, yields twoestimates for Zsd:

${{Z_{SD}\left( {\alpha = {1.5 \times 10^{6}\frac{1}{\sec}}} \right)} = {{\frac{98.6}{1 + \left( {0.919 - 1} \right)}\left( {1 - \left( {0.919 - 1} \right)} \right)} = {116\Omega}}},\mspace{14mu}{or}$${{Z_{SD}\left( {\alpha = {1.5 \times 10^{6}\frac{1}{\sec}}} \right)} = {{\frac{49.5}{1 + \left( {0.598 - 1} \right)}\left( {1 - \left( {0.598 - 1} \right)} \right)} = {116\Omega}}},$for this T1 cable.

This process was repeated for α's from 5×10⁵ sec⁻¹ to 10×10⁶ sec⁻¹ andthe results are shown in FIG. 16.

Example 4 Precision Measurement of Long Transmission Line Length orDistance to an Impedance Change Using a Reflected Wave

This example describes an alternate precision TOF measurement method ofSD test pulses for higher loss, longer lines whose attenuation of thetest pulses is so extensive that their amplitudes become too small asthey travel down the line for a common threshold crossing measurement tobe feasible.

The time of flight measured at a constant voltage threshold becomes moredifficult to estimate as the waveform is attenuated. This can be seen byexamining the voltage traces along an approximately 6-kft, 24 AWG, fiftytwisted wire pair, telephone cable (See FIG. 17). Three pairs of thefifty have been connected in series to generate an approximately 18-kftcable with three equal length sections. The applied SD input waveform isDe^(αt), with D of 0.19 volts and an α of 3.5×10⁵ l/sec. This α isapproximately one tenth of the α used for the 2-kft T1 cable. Thesmaller α is required here to reduce the attenuation of the truncated SDleading edge of the propagating pulse over these longer distances asdiscussed in section two. This allows signal detection over longerdistances. The value of A_(SD) and v_(SD) for this cable and α weredetermined using the procedure of example 1. These measured values arebased on assuming a 6000 ft calibration length for each twisted pairlength inside the telephone cable. The results are v_(SD)=4.348×10⁸ft/sec and A_(SD)=8.279×10⁻⁴ l/ft.

Even though the signal was detectable at much greater distances than 6kft, the constant threshold time of flight could not be measured. FIG.17 shows the voltage traces measured at the input, ˜6 kft, ˜12 kft, and˜18 kft distances, and it also shows the preservation of shape of the SDregion of the highly attenuated propagating waveform. The 18 kft traceis corrected for the reflection from the open termination by dividingthe measured waveform in half. The last two traces show that even withthis smaller α, beyond 12,000 ft there is little or no SD signal above0.19 volts. This is the value of the input SD waveform parameter D, thusthere is little or no SD voltage threshold common to the input SDwaveform and the SD waveforms at these longer distances.

The first step in precisely measuring highly attenuated SD pulse time offlight is to detect the SD region of the propagating pulse, V(t),measured and recorded at the distance #along the cable. This is done bycomputing the ratio

$\frac{{\mathbb{d}{V(t)}}/{\mathbb{d}t}}{V(t)}$of the measured pulse waveform. In the SD region of the pulse, thisratio is α. The end of the SD region is then found by detecting the timethat this ratio diverges from α. The ratio

$\frac{{\mathbb{d}^{2}{V(t)}}/{\mathbb{d}t^{2}}}{{\mathbb{d}{V(t)}}/{\mathbb{d}t}}$can also be used to detect the end of the SD region. In the SD region,this ratio is also α. The later ratio of the second and first derivativeresponds more quickly, but is also more susceptible to noise than theratio of the first derivative and the signal. These ratios as a functionof time are plotted in FIG. 18 for the SD waveform measured at 12-kft.In general, the ratio of

$\frac{\frac{\partial^{n}{V\left( {x,t} \right)}}{\partial^{n}t}}{\frac{\partial^{n - 1}{V\left( {x,t} \right)}}{\partial^{n - 1}t}}$for any positive n will have this property. Additionally, the ratio ofthe signal and its integral can be used to locate the SD region. In theSD region, we have the relationship:

$\frac{{De}^{\alpha\; t}}{\int_{0}^{t}{{De}^{\alpha\; t}{\mathbb{d}\tau}}} = {\frac{{De}^{\alpha\; t}}{\frac{1}{\alpha}\left( {{De}^{\alpha\; t} - D} \right)} = {\alpha{\frac{1}{1 - {\mathbb{e}}^{{- \alpha}\; t}}.}}}$Thus, the ratio will converge on α as time progresses. This ratiodiverges at the end of the SD region. In a lossy transmission mediumwith constant transmission line parameters, the end of the estimated SDregion can provide a good marker to determine the speed of light in themedium. The truncated SD leading edge of the pulse does not disperse,even with frequency dependent parameters. The high frequency componentsof the closing pulse, however, do undergo dispersion. These fast, highfrequency components tend to erode the end of the truncated SD region asthe pulse propagates. At long distances, these high frequency componentsare also more highly attenuated so the amount of SD region erosion isnot proportional to the distance traveled. In this case, the detectedend of the SD region serves as an initial estimate of the pulse time offlight or the distance the wave has traveled and serves to define theregion examined for more precise time of flight measurements in the nextstep.

The attenuation of the truncated SD signal propagating a distance d wasshown to be e^(−A) ^(sd) ^(·(1−v) ^(sd) ^(·√{square root over (LC)})d).In the application of time domain reflectometry (TDR) to these lines,the parameter 1/√{square root over (LC)} is commonly referred to as Vpand is given as a fraction of the speed of light in a vacuum. Thestandard value of Vp used in TDR for this air core poly 24 AWG cable is0.67. However, the value of the ratio used in this analysis is 0.59. Thereason for this can be seen in FIG. 19. The value of 0.67 is appropriatefor the high frequencies that are found in a standard TDR pulse used totest these telephone lines, but the α selected here for low truncated SDsignal attenuation in these long lines is equivalent to a low frequencyand propagates at a slower speed.

Once the initial TOF estimate of end of the SD region in the leadingedge of the propagating pulse is obtained at the distance l, the inputwaveform is attenuated and time shifted assuming the end of the SDregion of the traveling wave corresponds to the point of truncation ofthe initial SD waveform, (FIG. 20, top). The input waveform is thenincrementally translated in time and attenuated based on each newestimated distance traveled. At each distance estimate, this shifted andattenuated input waveform is subtracted from the measured waveform. Ifthere is no dc voltage offset and no noise, this difference will be zeroin the SD region when the estimated distance is equal to the actualdistance. Because there is always noise and there is frequently avarying differential, dc offset at different points along the cable, thevariance of this difference is evaluated at each estimated distance.This correlation has a minimum when the input waveform has beentranslated to the actual distance traveled, (FIG. 20, middle). If thereis a dc differential offset between the lines, this minimum will benon-zero. At this estimated distance, the erosion of the end of thetruncated SD region due to the dispersion from the closing pulse isobvious. If the waveform is translated too far, this deviation errormeasurement increases, (FIG. 20, bottom). FIG. 21 shows the calculatedstandard deviation of the difference of the shifted and attenuated inputwaveform and the SD waveform measured at the splice between the secondand third twisted wire pairs as a function of estimated distance. Theminimum occurs at an estimated distance of 12,216 ft. FIG. 22A shows thedifference of the two waveforms and the variance of the difference ofthe waveforms at the minimum. FIG. 22A indicates that there is a five toten millivolt differential dc offset between the two measuring points.FIG. 22B is presented to show how the variance has increased when theestimated distance is 38 ft longer than the distance associated with theminimum.

This process was repeated eight times giving average distances of 6,113ft, 12,216 ft, and 18,327 ft for the best-fit shift locations of theattenuated input waveform to the measured waveforms. The standarddeviations of the three lengths were 1 ft, 1 ft, and 5 ft respectively.The fits associated with one of these data sets are shown in FIG. 23.The actual length of the cable was not exactly known, however, each pairin the cable is parallel and the only differences in length are smalland due to different twist rates. This process yields the distance oftwo lengths within 10 ft of two times the distance of one length. Thedistance of three lengths is within 11 ft of three times the distance ofone length. Using the ˜6 kft distance as a standard gives a discrepancyof −0.08% for the ˜12 kft length and −0.06% for the ˜18 kft length.

Example 5 Time Domain Reflectometer (TDR) Precision Measurement ofTransmission Line Distance to Impedance Change Using a Reflected Wave

A Time Domain Reflectometer (TDR) is a test instrument used to findfaults in transmission lines and to empirically estimate transmissionline lengths and other parameters characterizing the line such asinductance per unit length, capacitance per unit length, resistance perunit length and conductance per unit length. A fundamental measurementin TDR test technology is the time of flight (TOF) of a test pulsegenerated by the instrument and applied to the line. This time of flightmay be measured by timing the passage of the pulse detected at twolocations along the line, referred to as Time Domain Transmissionmeasurements (TDT). Or by Time Domain Reflection measurements whichestimate the launch time of a pulse at one position and the subsequentreturn time of the pulse back to the same location after the pulse hasbeen reflected from a fault or other impedance change some distancealong the line. These measured TOF values, along with a value of thepropagation speed of the pulse, allows one to obtain the distancebetween measurement points or in the case of the reflected wave, thedistance from the pulse launch point to the location of the impedancechange causing the pulse to be reflected and returned to the launchpoint.

A fundamental limitation in TDR technology is the poor accuracy of theseTOF measurements in lossy, dispersive transmission lines. The relativelyhigh TDR accuracy of TOF values obtainable in short low loss, lowdispersion transmissions lines is possible only because the propagatingtest pulses keep their shape and amplitude in tact over the distancesthey travel during TOF measurements. By contrast, in dispersive, lossylong transmission lines the test pulses used in the art change shape,change amplitude any speed as they travel. The TOF measurements used inthe art under these circumstances focus on estimating the emergence timeof the leading profile of the test pulses. This part of the signature oftest pulses used in the art has characteristically a low signal leveland low signal slope making an accurate pulse emergence time measurementdifficult to obtain.

Several advantages can be obtained in TDR technology for lossy,dispersive transmission lines by using a test pulse that contains atruncated SD signal in its leading edge. This truncated SD leading edgetravels at a constant speed along these transmission lines withoutchanging shape. The speed of propagation of this SD edge is a functionof the line parameters and the SD signal parameter alpha and iscontrollable by changing alpha. The truncated SD leading edge of thetest pulse will be attenuated as it travels along a lossy line. However,this rate of attenuation is also a function of the line parameters andthe SD signal parameter alpha and is controllable by changing alpha.

The same principles used for TDR can be applied to waveforms other thanelectromagnetic waves in a transmission line. The reflections ofacoustic waves in SONAR or geophysical applications can be analyzedusing the techniques discussed in this section to provide accurate timeof flight and distance estimates as well as be used to characterize thetransmission media.

The accuracy of the SD test waveform in a TDR application can bedemonstrated by an example using the T1 cable, as discussed in examples1 and 2. The two twisted wire pairs inside the T1 cable are connected inseries to form a 2004 ft cable and the truncated SD signal is applied tothe input. The voltage trace is measured at the splice of the two linesat 1002 ft. This can eliminate the need to correct for any lineimpedance mismatch at the point of measurement. The exponentialcoefficient α of the applied wave is 6.7×10⁶ sec⁻¹. The SD parameters,A_(SD) and v_(SD), are obtained from the empirical transfer functionanalysis presented in example 2. Interpolating from the data of table Igives an A_(SD) of 10.703×10⁻³ ft⁻¹ and a v_(SD) of 6.276×10⁸ ft/sec forthis α.

The first demonstration terminates the cable with an open circuit. Thereflected coefficient of an open circuit is +1, which results in apositive reflection. FIG. 24 shows the voltage waveform measured at the1002 ft point showing the forward traveling applied wave and the wavereflected from an open termination at the end of the cable. FIG. 25shows the result of the measured reflected wave together with theapplied wave time shifted and attenuated according to the procedure ofexample 4. The total distance traveled by the reflected wave is measuredto be 2004 ft by this process, or exactly two times the 1002 ft lengthof this twisted wire pair section. FIGS. 26A-B show the constantthreshold timing method result for TOF discussed in example 1. Thismethod results in a time of flight of 3,203 nsec, which gives anestimated total distance traveled of 2010 ft. This results in anestimated twisted wire pair section length of 1005 ft, or 3 ft in error(+0.3%). This is an illustration of reduced accuracy using the constantthreshold TOF measurements when the pulse travels far enough toexperience significant attenuation. The accuracy of this method maydecline more in cases where the test pulse experiences even moreattenuation. This level of attenuation may not generally occur indigital circuits. The maximum attenuation of digital signals ininterconnects is generally less than one-half of the driving signal. Insuch cases, the constant threshold TOF measurements with the SD waveshould be preferably more accurate than current TDR methods usingestimated time of emergence of the reflected pulses.

The next demonstration terminates the second twisted wire pair with ashort circuit. The reflected coefficient at a short circuit is −1, whichresults in a negative reflection. FIG. 27 shows the voltage waveformmeasured at the 1002 ft point showing the forward traveling applied waveand the wave reflected from a short circuit termination at the end ofthe second twisted wire pair. FIG. 28 shows the result of the invertedreflected wave and the applied wave time shifted and attenuatedaccording to the procedure of example 4. The total distance traveled ismeasured to be 2004 ft by this process, or exactly two times the 1002 fttwisted wire pair length. FIGS. 29A-B show the constant threshold timingmethod of estimating TOF discussed in example 1. This method results ina time of flight of 3,200 nsec, which gives an estimated total distancetraveled of 2008 ft. This results in an estimated twisted wire pairsection length of 1004 ft, or 2 ft in error (+0.2%).

The utility of the SD waveform to the TDR process can be increased byusing an empirical transfer function obtained with the use of a non-SDpulse applied to the line in a way similar to the process discussed inexample 2. As an example with this cable, an experimental voltage pulseis generated by an arbitrary function generator connected to a singleunterminated twisted wire pair from the T1 cable. This waveform is shownin FIGS. 30-31 (dotted). The same waveform can be applied by the signalgenerator to a variable resistor load substituted for the 1002 fttwisted wire pair. The resistor is adjusted until the waveformapproximates the shape and magnitude of the waveform applied by thearbitrary waveform generator to the twisted wire pair, see FIGS. 30-31(solid). The waveform match is not exact because the resistor does nothave the exact same frequency response as that of the twisted wire pair.A load resistance equal to the characteristic impedance, Z_(o), of thetwisted wire pair results in a waveform that is closely matched to thewaveform applied to the cable and has similar frequency content. Thedifference between the signal measured with the resistive load and thesignal measured with the transmission line consists of the response ofthe transmission line to the applied signal and contains any reflectedwaves from impedance mismatches on the transmission line. The transferfunction is then calculated from the ratio of the fast Fouriertransforms (FFT) of the difference of the two signals, and the signal ofthe resistor alone:

${H_{est}({j\omega})} = {\frac{F\; F\;{T\left( {V_{{Measured}\mspace{14mu}{with}\mspace{14mu}{Cable}} - V_{{Measured}\mspace{14mu}{with}\mspace{14mu}{Load}\mspace{14mu}{Resistance}\mspace{14mu} Z_{o}}} \right)}}{F\; F\;{T\left( V_{{Measured}\mspace{14mu}{with}\mspace{14mu}{Load}\mspace{14mu}{Resistance}\mspace{14mu} Z_{o}} \right)}}.}$This function, shown in FIG. 32, can be used to evaluate the reflectedtransmission line response to a SD pulse.

As a test of this process, a SD pulse with an α equal to the α used inthe previous two experiments was simulated. The transfer functiondetermined for the open terminated T1 cable, (FIG. 32), was used tocalculate the expected response, see FIG. 33. There is a small pulse atthe time of the applied pulse due to the resistor and cable havingdifferent high frequency impedance values. The influence of this pulsewas eliminated with appropriate time windowing. The simulated responsewas then used to estimate the distance to the open termination using themethods of example 4 (FIG. 34), and example 1 (FIGS. 35A-B). The resultof the slide and attenuate method resulted in an estimated total roundtrip distance the pulse traveled of 2004, or double the exact distanceof 1002 ft to the open. The constant threshold method resulted in timeof flight of 3,196 nsec for an estimated distance traveled of 2006 ft,or two times an estimated distance of 1003 ft to the open (0.1% Error).

No actual SD signal was experimentally applied to the twisted wire pairto obtain these results. The test signal applied to the line (FIG. 30)was similar to the type obtained form a commercial TDR instrument fortesting communication cables with lengths ranging up to 5-kft. Themeasurements resulting from this test line voltage were used to obtainan empirical line transfer function. The SD TOF and the line distanceestimates were all obtained computationally by simulating the SDpropagation along the line utilizing this empirical line transferfunction estimated with non-SD signals.

Example 6 Communication by High Speed Data Transmission Using SpeedyDelivery Waveforms

The SD waveform has several properties that make it suitable as thebasis for a data transmission scheme. For example: the exponential shapeis maintained as it propagates at constant speed in a uniform cable; theattenuation of the SD waveform is adjustable by changing the exponentialwaveform parameter α; the propagating speed of the SD waveform isadjustable by changing the exponential waveform parameter α; and the SDwaveforms with different α's are linearly independent.

This example teaches using the first property as the basis for a methodof transmitting data using SD waveforms. The signals are transmitted2004 ft along a transmission line consisting of the two twisted wirepairs in a 1002 ft T1 cable connected in series. The second twisted wirepair is terminated with a 100-ohm resistor. The data is encoded on fourSD waveforms, each having a distinct α. Each waveform can be transmittedwith a positive or a negative D, resulting in eight total states forthree bits per symbol. One symbol is transmitted every threemicroseconds, giving a data rate of 1 Mbps. As one of ordinary skill inthe art will recognize with the benefit of this disclosure, severalother communication schemes can be readily derived from the teachingscontained herein.

The symbols can consist of a two-microsecond truncated SD leading edgeperiod followed by a one-microsecond recovery period. This recoveryperiod includes a variable width half-sine compensating pulse adjustedto shorten the time required for the transmission line to return to azero voltage. The need for this recovery period and compensating pulseis illustrated in FIGS. 36A-B and FIGS. 37A-B. FIG. 36A shows thetransmitted pulse without the compensating pulse, and FIG. 36B shows thereceived pulse without the compensating pulse. The truncated SD waveformin the leading edge of the input pulse is closed with a simple rampsignal, which is rounded due to the line response. Although the SDportion of the propagating pulse does not undergo deformation, theclosing ramp does disperse into the region beyond the three-microsecondsymbol period. FIGS. 37A-B show the effect of the compensating pulse ineliminating this inter-symbol interference. Each truncated SD waveformmay requires a different amount of compensation so the width of thehalf-sine pulse can be varied based on observing the pulse at theterminated end of the second twisted wire pair.

FIG. 38 shows a series of four symbols applied to the twisted wire pairinput. Each symbol has a unique α, and a unique D. The α's are equallyspaced 1.1×10⁶ sec⁻¹ apart at 2.2×10⁶ sec⁻¹, 3.3×10⁶ sec⁻¹, 4.4×10⁶sec⁻¹, and 5.5×10⁶ sec⁻¹. The order α's used for this illustration offour successive symbols in FIG. 9.3 is 4.4×10⁶ sec⁻¹, 2.2×10⁶ sec⁻¹,3.3×10⁶ sec⁻¹, and 5.5×10⁶ sec⁻¹. The D's were chosen to maintain anapproximately equal maximum and minimum symbol voltage. For thisdemonstration, the fourth waveform is inverted. The signal measured atthe 100-ohm termination at a distance of 2004-ft is shown in FIG. 39.FIG. 40 shows the natural log of the signal within each symbol periodfor the signal of FIG. 39. A set of matched filters could be used todetect the individual α's. The slopes of the least square linear fits ofthe linear regions of these waveforms are shown in Table II (set forthbelow). The decision level for this demonstration would be one half ofthe step size in α (1.1×10⁶ sec⁻¹) of 0.55×10⁶ sec⁻¹. All errors arewell below this threshold.

TABLE II Encoded and decoded α's from Data Transmission DemonstrationApplied α Detected α Error Symbol (×10⁶ sec⁻¹) (×10⁶ sec⁻¹) (×10⁶ sec⁻¹)1 4.4 4.38 −0.02 2 2.2 2.23 +0.03 3 3.3 3.28 −0.02 4 5.5 5.53 +0.03

Example 7 Communication by High Speed Data Transmission Using OrthogonalLinear Combinations of Speedy Delivery Waveforms

This example discusses using the fourth SD waveform property, that SDwaveforms with different α's are linearly independent. This property canbe utilized to generate an orthogonal set of pulses. Data can be encodedon these pulses by varying their amplitude. These amplitude-modulatedorthogonal pulses are transmitted simultaneously on the transmissionline. At the receiver, the individual pulse amplitudes are computed bytaking advantage of their orthogonality. The basic pulses derived fromthe SD waveforms must be orthogonal at the receiver for this scheme towork.

The repetitive transmission of data may require closed pulses. As shownin example 6, that the SD portion of the leading edge of a closedpropagating pulse maintains its shape, while the shape used to close thepulse will disperse as it propagates. The attenuation and propagationspeed of the truncated SD leading region of the pulse is a function ofthe SD parameter α. Therefore, the orthogonality of a set of thesepulses with a variety of α values will be reduced as the pulsespropagate. Fortunately, the pulses only have to be orthogonal at thereceiver. Techniques can be used to generate a set of orthogonal pulsesfrom linearly independent component pulses. The effect of thetransmission line on these component pulses can be determinedempirically. These transmitted component waveforms, measured at thereceiver, will be linearly independent. Linear combinations of thesemeasured, linearly independent component pulses are used to generate aset of pulses that are orthogonal at the receiver. The constantsdetermined by the orthogonalization process are supplied to thetransmitter and used to generate transmitted pulses that will beorthogonal at the receiver. An example of this procedure follows. Theprocess of generating and transmitting the orthogonal pulses issimulated for an 8 kft, 26 AWG twisted wire pair transmission lineterminated with 100 Ohms resistance.

The example orthogonalization process can begin by using a set of SDwaveforms X_(m)(t)=e^((α) ^(o) ^(+(m−1)·Δα)·t) for m=1 . . . 5,

${\alpha_{o} = {1 \times 10^{4}\frac{1}{\sec}}},{{\Delta\alpha} = {1 \times 10^{5}\frac{1}{\sec}}},$and t=0 . . . 7×10⁻⁶ sec. The five waveforms are shown for aseven-microsecond interval in FIG. 41. The waveforms are linearlyindependent on this time interval. Additionally, these SD waveforms havea polynomial structure, i.e. each waveform is equal to e^((Δα)·t) timesthe previous waveform. This allows the use of a simple recursivealgorithm for polynomials [4] to create orthonormal linear combinationsof these linearly independent waveforms. The set of five orthonormal Y'sresulting from this procedure is shown in FIG. 42. In this case, X and Yare related by:

$\begin{matrix}{Y = {\overset{\_}{A} \cdot X}} \\{\begin{bmatrix}Y_{1} \\Y_{2} \\Y_{3} \\Y_{4} \\Y_{5}\end{bmatrix} = {\begin{bmatrix}0.026 & 0 & 0 & 0 & 0 \\{- 0.129} & 0.088 & 0 & 0 & 0 \\0.717 & {- 1.003} & 0.337 & 0 & 0 \\{- 4.074} & 8.578 & {- 5.874} & 1.310 & 0 \\23.304 & {- 65.519} & 67.862 & {- 30.704} & 5.124\end{bmatrix} \cdot {\begin{bmatrix}X_{1} \\X_{2} \\X_{3} \\X_{4} \\X_{5}\end{bmatrix}.}}}\end{matrix}$

The set, Y, is orthonormal in that the inner product

${\left\langle {Y_{i},Y_{j}} \right\rangle \equiv {\int_{0}^{T = {12{\mu sec}}}{{{Y_{i}(t)} \cdot {Y_{j}(t)}}{\mathbb{d}t}}}} = \left\{ {\begin{matrix}0 & {i \neq j} \\1 & {i = j}\end{matrix}.} \right.$

The sharp waveform transitions at zero and seven microseconds have highfrequency content that would disperse during transmission and contributeto inter-symbol interference. This effect is reduced by multiplying thisorthonormal set by sin(ω·t), with

$\omega = {\frac{\pi}{7 \times 10^{- 6}}.}$This has the effect of changing the SD waveform to X′, where

${X_{m}^{\prime}(t)} = {{{\mathbb{e}}^{{({\alpha_{o} + {{({m - 1})}{\Delta\alpha}}})} \cdot t} \cdot {\sin\left( {\omega\; t} \right)}} = {\frac{1}{2j}\begin{bmatrix}{{\mathbb{e}}^{\lbrack{{({\alpha_{o} + {{({m - 1})}\Delta\;\alpha}})} + {j\omega}}\rbrack} -} \\{\mathbb{e}}^{\lbrack{{({\alpha_{o} + {{({m - 1})}\Delta\;\alpha}})} - {j\omega}}\rbrack}\end{bmatrix}}}$for m=1 . . . 5.

Although the SD waveforms, X′, are still linearly independent, theprevious linear combination of these waveforms, Y′=Ā·X′, is no longerorthogonal. The waveforms, Y′, are shown in FIG. 43. These waveforms maybe transmitted through the channel to the receiver. At the receiver, thewaveforms show the trailing effects of dispersion beyond the 12 μsecsymbol period, see FIG. 44. If the spread of the pulses into thefollowing symbol periods is not removed, this inter-symbol interferenceat the receiver will render data encoded at the transmitter unreadable.A simple method of eliminating this interference is by adding a pulsewith a compensating tail, Z=Y′+b·C. In this example, the compensationpulse, C(t), is one cycle of

$1 - {\cos\left( {\frac{\pi}{2.9 \times 10^{- 6}} \cdot t} \right)}$delayed one microsecond after the start of the symbol period, see FIG.45. FIGS. 46, 47 show the waveforms at the input and at 8 kft withappropriately sized compensating pulses added to each signal. Theduration of these compensated component pulses at the receiver are nowshorter than the desired 12 μsec symbol period (see FIG. 47), so anylinear combination of these pulses will not interfere with the nextsymbol.

The multiplication by the sine pulse, the adding of the compensatingpulse, and the transmission in the channel, have all reduced theorthogonality of the initial five pulses. Initially the angles betweenthe pulses in FIG. 42 were all 90°. Now the angles between the pulses inFIG. 47 range from 11° to 74°. These five linearly independent waveformsare now combined into set, S, of four orthonormal waveforms usingprinciple component decomposition (PCD) [5]. The Gram-Schmidt (G-S)orthogonalization method [6] may also be used for this step in theprocedure, however the PCD method is preferable. The G-S method expandsan orthonormal vector set one vector at a time. This can lead to apoorly conditioned transformation matrix if the components are close tobeing linearly dependent. The PCD method is preferable because it takesinto account the entire inner-product space spanned by all of thecomponents. The process makes use of the eigenvectors of the entireinner-product space to generate the transformation matrix. Additionally,the transformation matrix can be easily manipulated to order theeigenvalues. In this example the four largest eigenvalues were selectedby truncating the transformation matrix. This process yields atransformation matrix, B, that generates four waveforms that areorthonormal at 8-kft by linearly combining the five component signals.These waveforms are shown in FIG. 48 (the transmitted signals) and FIG.49 (the received orthogonal signals).

$\begin{matrix}{S = {\overset{\_}{B} \cdot Z}} \\{S = {\overset{\_}{B} \cdot \left( {Y^{\prime} + {b \cdot C}} \right)}} \\{\begin{bmatrix}{S\; 1} \\{S\; 2} \\{S\; 3} \\{S\; 4}\end{bmatrix} = {\begin{bmatrix}4.69 & 5.75 & {- 21.05} & {- 41.44} & 46.26 \\3.55 & 0.41 & 32.53 & {- 33.72} & 6.00 \\{- 2.74} & 9.11 & 0.48 & {- 63.36} & {- 91.60} \\{- 3.35} & {- 3.35} & 2.06 & {- 56.60} & 123.38\end{bmatrix} \cdot \begin{pmatrix}{\begin{bmatrix}Y_{1}^{\prime} \\Y_{2}^{\prime} \\Y_{3}^{\prime} \\Y_{4}^{\prime} \\Y_{5}^{\prime}\end{bmatrix} +} \\{\begin{bmatrix}{- 0.04034} \\{- 0.00283} \\0.01971 \\{- 0.00007} \\{- 0.00110}\end{bmatrix} \cdot C}\end{pmatrix}}} \\{\begin{bmatrix}{S\; 1} \\{S\; 2} \\{S\; 3} \\{S\; 4}\end{bmatrix} = {{\begin{bmatrix}{1,231.1} & {{- 3},365.0} & {3,375.8} & {{- 1},474.7} & 237.0 \\300.5 & {- 714.9} & 616.1 & {- 228.4} & 30.7 \\{{- 1},877.3} & {5,458.0} & {{- 5},843.5} & {2,729.4} & {- 469.3} \\{3,107.7} & {{- 8},571.5} & {8,705.9} & {{- 3},862.4} & 632.2\end{bmatrix} \cdot \begin{bmatrix}{\mathbb{e}}^{1 \times 10^{4}t} \\{\mathbb{e}}^{1.1 \times 10^{5}t} \\{\mathbb{e}}^{2.1 \times 10^{5}t} \\{\mathbb{e}}^{3.1 \times 10^{5}t} \\{\mathbb{e}}^{4.1 \times 10^{5}t}\end{bmatrix} \cdot {\sin({\omega t})}} + {\begin{bmatrix}{- 0.668} \\0.493 \\0.199 \\0.054\end{bmatrix} \cdot C}}}\end{matrix}$

The orthonormal set S is orthogonal and normalized at the receiver. Theprocess accounts for the effects of the channel and allows thetransmitter input waveforms to be easily constructed as a linearcombination of defined components prior to transmission. This set S canbe used as the basis functions for an orthogonal pulse amplitudemodulation data transmission scheme. This process is depicted in FIG.50.

The data can be encoded by amplitude modulation with five bits on eachof the four orthonormal pulses, S. Five bits requires thirty-two states.Each state corresponds to one amplitude level, a_(i), on each orthogonalpulse, S_(i). For this example the amplitude levels were ±0.5, ±1.5, . .. ±15.5. These four modulated orthogonal pulses are summed to generate asymbol, Q=a₁S₁+a₂S₂+a₃S₃+a₄S₄.

The symbol is transmitted and decoded at the receiver by using theorthogonality of the pulses. For example, the applied signal onorthogonal pulse one can be found by taking the inner product of thesymbol, Q, with the pulse one signal, S₁. This is done by multiplyingthe received symbol by the known orthogonal pulse, integrating over thesymbol period, and normalizing the result.

$\begin{matrix}{\frac{\left\langle {Q,S_{1}} \right\rangle}{\left\langle {S_{1},S_{1}} \right\rangle} = \frac{\left\langle {{{a_{1}S_{1}} + {a_{2}S_{2}} + {a_{3}S_{3}} + {a_{4}S_{4}}},S_{1}} \right\rangle}{\left\langle {S_{1},S_{1}} \right\rangle}} \\{= \frac{\left\langle {{a_{1}S_{1}},S_{1}} \right\rangle + \left\langle {{a_{2}S_{2}},S_{1}} \right\rangle + \left\langle {{a_{3}S_{3}},S_{1}} \right\rangle + \left\langle {{a_{4}S_{4}},S_{1}} \right\rangle}{\left\langle {S_{1},S_{1}} \right\rangle}} \\{= \frac{{a_{1}\left\langle {S_{1},S_{1}} \right\rangle} + 0 + 0 + 0}{\left\langle {S_{1},S_{1}} \right\rangle}} \\{= a_{1}}\end{matrix}$

FIG. 51 shows three successive symbols at the output of the transmitterand FIG. 52 shows the same three symbols at the input of the receiver.The symbols were encoded with a random selection of five bits perorthogonal pulse for a data rate of

$\frac{4\frac{Pulses}{Symbol} \times 5\frac{Bits}{Pulse}}{12 \times 10^{- 6}\frac{\sec}{Symbol}} = {1.67\mspace{14mu}{{Mbps}.}}$

FIG. 53 is a histogram of the error in detected level,a_(expected)−a_(detected), for one second, or 1.67×10⁶ bits, of datatransmitted with −140 dBm additive white Gaussian noise (AWGN) present[7]. The standard deviations of the error distributions are 0.016,0.018, 0.020, and 0.018 with maximum detected errors of 0.074, 0.071,0.083, and 0.078. The decoding amplitude decision level is ±0.5

Example 8 On-Chip Clock Circuits

This example teaches the use of SD waveforms in digital circuits,specifically, in on-chip clock circuits. On-chip clock circuits maybehave like transmission lines with the increased clock rates.

A major design advantage of the SD waveforms when used as clock signalsis that the delay in an RC line (√{square root over (RC/α)}·l) is linearwith length, l, instead of quadratic with length as it is with thecommonly assumed step signal. Linear delay with length

$\left( {\sqrt{LC}{\sqrt{1 + \frac{R}{L\;\alpha}} \cdot l}} \right)$is also true for the SD signal in an RLC line. In contrast, the delay inRLC transmission lines excited by a conventional signal exhibit anexponentially growing increase in delay with length [8].

The fact that the SD signal delay increases as a linear function oflength for all types of clock lines can simplify the clock line layoutdesign process. This linear relationship holds for the entire range ofclock line behaviors, from the very lossy clock lines that behave likeRC transmission lines to low loss clock lines that behave like RLCtransmission lines. This simple linear relationship between line lengthand SD clock signal delay provides a basis for implementing software CADtools that may enhance accuracy while reducing design computationalrequirements.

Time Skew Clock control is a major issue limiting system performance. Asolution to this problem includes equalizing conductor lengths to thevarious locations on the chip where the clock signal is needed. Severalgeometrical patterns are commonly used (H, etc.) to equalize the signalpath lengths, thereby equalizing signal delays. This requires equalizingthe path lengths along the traces delivering the clock signal to pinlocations that are physically close to the master chip clock driver withthose clock signal path lengths to pin locations at the longestdistances from the master driver. Small adjustments to equalize thedelays may be accomplished by adjusting the line widths to modify theline parameters and thereby the delay per unit length and alternativelyby active means such as varying the delay of delayed-locked loopsembedded in the clock lines.

The invention provides a method and/or apparatus for utilizing truncatedSD clock waveforms to reduce the clock skew. An advantage of theinvention is that the delay per unit length can be adjusted with theexponential coefficient, α, chosen in the clock line driver design. Thiscontrollable delay per unit length has been demonstrated for a 100-ftcoaxial cable in the initial patent application. The minimum coaxialcable delay measured was increased by a factor of 1.8.

The large increases in the clock signal delay per unit length achievedby varying the exponential coefficient α allows lines to the pins closeto the clock driver to be shortened to more closely approximate theminimum physical distance between the driver and pin while maintaining apath delay equal to the delay of the longest clock lines on the chip.The shorter clock distribution lines may utilize less physical space onthe chip and reduce the total clock line capacitance. This reduction inclock line capacitance reduces the total power consumption required bythe chip for a given clock frequency. In one embodiment, an SD waveformdriver may also be made to create an SD clock signal with adjustabledelay by including a mechanism for adjusting the exponential coefficientα in the clock generator. This adjustable delay of the SD clock signalmay provide a larger delay adjustment range than the delayed-lockedloops utilized in the current technology.

Example 9 Signal Propagation Without Distortion in a Dispersive LossyMedia

The propagation of signals in a no-loss non-dispersive media is idealbecause there is no distortion.^([9]) However, pulse shapes thatpropagate along lossy, dispersive transmission lines are commonlydispersive, i.e., change shape^([10]). This property can lead toinaccuracy of measurements and mischaracterizations of propagationbehaviors.

As noted in Example 4 and Example 5 and illustrated in FIG. 17, thepropagated waveform (e.g., a speedy delivery waveform) maintains anessentially constant shape during transmission in a dispersive lossymedia. In particular, a speedy delivery (SD) waveform can be shown to bean exact closed form solution in the time domain of the four-parameterTelegrapher's equation that can be expressed as F(vt−x). Thetelegrapher's equation for the voltage on a transmission line is^([11])

$\begin{matrix}{{{{LC}\frac{\partial^{2}v}{\partial t^{2}}} + {\left( {{LG} + {RC}} \right)\frac{\partial v}{\partial t}} + {RGv}} = \frac{\partial^{2}v}{\partial x^{2}}} & {{Eq}.\mspace{14mu} 1}\end{matrix}$where L, C, R, and G are the transmission line parameters^([12]).

The SD solution of Eq. 1 in the time domain with boundary conditionsv(x=0,t)=De^(αt) and v(x→∞,t)=0 isv _(SD)(x,t)=De ^(αt) e ^(−xγ)(α)  Eq. 2which, when rewritten in the form F(vt−x) yields,v _(SD)(x,t)=De ^(γ(α)[(v) ^(SD) ^((α))t−x])  Eq. 3where γ²(s)=LCs²+(LG+RC)s+RG and

${v_{SD}(\alpha)} = {\frac{\alpha}{\sqrt{{{LC}\;\alpha^{2}} + {\left( {{GL} + {RC}} \right)\alpha} + {RG}}}.}$Thus, the SD wave does not change shape as it propagates with constantvelocity on this dispersive lossy transmission line. The SD solution'spropagation velocity depends on the SD boundary condition waveformparameter α and the principal line parameters L, C, R, and G (thedependence on α is removed if R=0 and G=0, in which case v=1/√{squareroot over (LC)}, the standard result for the ideal transmission line).

If the principal transmission line parameters are frequency dependent,the SD steady state solution is still valid, where the signalpropagation velocity becomes

$\begin{matrix}{v_{SD} = \frac{\alpha}{\sqrt{{{\overset{\_}{L}(\alpha)}{\overset{\_}{C}(\alpha)}\alpha^{2}} + {\left( {{{\overset{\_}{G}(\alpha)}{\overset{\_}{L}(\alpha)}} + {{\overset{\_}{R}(\alpha)}{\overset{\_}{C}(\alpha)}}} \right)\alpha} + {{\overset{\_}{R}(\alpha)}{\overset{\_}{G}(\alpha)}}}}} & {{Eq}.\mspace{14mu} 4}\end{matrix}$where L(s), C(s), R(s), and G(s) are the Laplace transforms of theprincipal line parameters. As such, the SD wave represented by Eq. 2does not change shape as it propagates with constant velocity on thisdispersive lossy transmission line with frequency dependent parameters.

Any application of the SD signal in a circuit can require that theexponential signal be truncated at some maximum amplitude. The result oftruncating the amplitude of the SD signal at the line input is that theresponses measured at different locations along the transmission lineare also truncated SD signals having the same shape, but whose peakamplitudes decline with distance. Formulas for this attenuation in thetruncated SD signal amplitude and the propagation velocity of thetruncated SD signal are contained in references for various types oftransmission lines and other DLM. It is also noted that using atruncated SD signal as the forward edge of a closed pulse and closingthe pulse with a non-SD waveform results in dispersion of the pulse waveform as a whole, with only the leading SD edge retaining the shape ofthe waveform as it propagates in DLM.

Example 10 Propagation of a Truncated Exponential Waveform Modulatedwith a Sinusoidal Carrier

In this example, an electrical signal which includes a truncated SDsignal envelope modulated with a sinusoidal carrier signal can propagateon a lossy dispersive RLC transmission line without distortion isdemonstrated. Similarly, electromagnetic plane waves which can include atruncated SD signal envelope modulated by a sinusoidal electromagneticplane wave propagating through a dispersive plasma media (e.g., anionosphere) is also demonstrated. Furthermore, the example illustratesthat the propagation speed and attenuation of the envelope can becontrolled.

1. Steady State Response of a Lossy DispersiveResistance-Capacitance-Inductance Transmission Line to a Truncated SDSignal Envelope Modulated by a Sinusoidal Carrier.

The propagation transfer function for aresistance-capacitance-inductance (RLC) transmission line is^([12])V(z,s)=B(s)e ^(−zγ(s)) or z>0  Eq. 5where^([12]) γ²(s)=LCs²+RCs and the SD boundary condition isb(t)=V(t,z=0)=De^(αt) sin ω₀t. Note that the boundary condition is a SDenvelope modulated with a carrier of frequency ω₀ that is understood tobe truncated in time. As such, the steady state solution in time domainis

$\begin{matrix}{{V\left( {t,z} \right)} = {{\frac{D}{2j}{\mathbb{e}}^{\alpha\; t}{\mathbb{e}}^{{j\omega}_{0}t}{\mathbb{e}}^{{- z}\;{\gamma{({\alpha + {j\omega}_{0}})}}}} - {\frac{D}{2j}{\mathbb{e}}^{\alpha\; t}{\mathbb{e}}^{{- {j\omega}_{0}}t}{\mathbb{e}}^{{- z}\;{\gamma{({\alpha - {j\omega}_{0}})}}}}}} & {{Eq}.\mspace{14mu} 6}\end{matrix}$Defining P(α,ω₀)=LCα²−LCω₀ ²+RCα>0 if LCα²+RCα>LCω₀ ² andQ(α,ω₀)=2LCαω₀+RCω₀>0, then Eq. 6 becomes

$\begin{matrix}{{V\left( {t,z} \right)} = {D\;{\mathbb{e}}^{{\alpha\; t} - {{z{({P^{2} + Q^{2}})}}^{1/4}{\cos(\frac{\tan^{- 1}{({Q/P})}}{2})}}}{\sin\left\lbrack {{\omega_{0}t} - {{z\left( {P^{2} + Q^{2}} \right)}^{1/4}{\sin\left( \frac{\tan^{- 1}\left( {Q/P} \right)}{2} \right)}}} \right\rbrack}}} & {{Eq}.\mspace{14mu} 7}\end{matrix}$Thus, the envelope of the carrier is

$\begin{matrix}{D\;{\mathbb{e}}^{{\lbrack{{({P^{2} + Q^{2}})}^{1/4}{\cos{(\frac{\tan^{- 1}{({Q/P})}}{2})}}}\rbrack} \times {\{{{{({\alpha\; t})}/{\lbrack{{({P^{2} + Q^{2}})}^{1/4}{\cos{(\frac{\tan^{- 1}{({Q/P})}}{2})}}}\rbrack}} - z}\}}}} & {{Eq}.\mspace{14mu} 8}\end{matrix}$which has the form of a function F(vt−z) which propagates with acontrollable velocity v and controllable attenuation of a truncated SDenvelope without changing shape^([12]). Therefore, the propagationvelocity of the envelope is:

$\begin{matrix}{v = \frac{\alpha}{\left( {P^{2} + Q^{2}} \right)^{1/4}{\cos\left( \frac{\tan^{- 1}\left( {Q/P} \right)}{2} \right)}}} & {{Eq}.\mspace{14mu} 9}\end{matrix}$

2. One Dimensional Electromagnetic Plane Wave Propagation in aDispersive Media^([3])

Assuming the electromagnetic field components are only a function of zand assuming the E-field is polarized in the x-direction, theelectromagnetic field components are as follows:Ē={circumflex over (x)}E _(x)(z,t)  Eq. 10H=ŷH _(y)(z,t)  Eq. 11J={circumflex over (x)}J _(x)(z,t)  Eq. 12ω_(p) ²=ω_(p) ²(z,t)  Eq. 13If ω_(p) ²(z,t)=ω_(p) ² (i.e., assuming no variations in the plasmafrequency in z direction or time), then

${\frac{\partial^{2}E}{\partial z^{2}} - {\frac{1}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}}} - {\frac{1}{c^{2}}{\omega_{p}^{2}\left( {z,t} \right)}E}} = {0^{\lbrack 10\rbrack}\mspace{14mu}{and}}$${\frac{\partial^{2}\overset{.}{H}}{\partial z^{2}} - {\frac{1}{c^{2}}\frac{\partial^{2}\overset{.}{H}}{\partial t^{2}}} - {\frac{1}{c^{2}}{\omega_{p}^{2}\left( {z,t} \right)}\overset{.}{H}} + {ɛ_{0}\frac{\partial}{\partial z}{\omega_{p}^{2}\left( {z,t} \right)}E}} = 0^{\lbrack 10\rbrack}$becomes

$\begin{matrix}{{\frac{\partial^{2}E_{x}}{\partial z^{2}} - {\frac{1}{c^{2}}\frac{\partial^{2}E_{x}}{\partial t^{2}}} - {\frac{1}{c^{2}}{\omega_{p}^{2}\left( {z,t} \right)}E_{x}}} = 0} & {{Eq}.\mspace{14mu} 14} \\{{\frac{\partial^{2}{\overset{.}{H}}_{y}}{\partial z^{2}} - {\frac{1}{c^{2}}\frac{\partial^{2}{\overset{.}{H}}_{y}}{\partial t^{2}}} - {\frac{1}{c^{2}}{\omega_{p}^{2}\left( {z,t} \right)}{\overset{.}{H}}_{y}}} = 0} & {{Eq}.\mspace{14mu} 15}\end{matrix}$Taking the Laplace Transform of the Eq. 14 of the E-field yields:

$\begin{matrix}{{\frac{1}{c^{2}}\left( {s^{2} + \omega_{p}^{2}} \right){{\overset{\_}{E}}_{x}\left( {s,z} \right)}} = \frac{\mathbb{d}^{2}{{\overset{\_}{E}}_{x}\left( {s,z} \right)}}{\mathbb{d}z^{2}}} & {{Eq}.\mspace{14mu} 16}\end{matrix}$Then the propagation transfer function for the E-field, in terms of theelectromagnetic boundary condition at the edge of the plasma, B(s), ise^(−γ(s)z) where

${{\gamma^{2}(s)} = {\frac{1}{c^{2}}\left( {s^{2} + \omega_{p}^{2}} \right)}},$and the SD boundary conditionb(t)=E(t,z=0)=De ^(αt) sin ω₀ t.  Eq. 17It is noted that the boundary condition is a SD envelope modulated witha carrier of frequency ω₀ that is understood to be truncated in time.

As such, the steady state solution in the time domain is

$\begin{matrix}{{E_{x}\left( {t,z} \right)} = {{\frac{D}{2j}{\mathbb{e}}^{\alpha\; t}{\mathbb{e}}^{j\;\omega_{0}t}{\mathbb{e}}^{{- z}\;{\gamma{({\alpha + {j\omega}_{0}})}}}} - {\frac{D}{2j}{\mathbb{e}}^{\alpha\; t}{\mathbb{e}}^{{- j}\;\omega_{0}t}{\mathbb{e}}^{{- z}\;{\gamma{({\alpha - {j\omega}_{0}})}}}}}} & {{Eq}.\mspace{14mu} 18}\end{matrix}$Defining P(α,ω₀)≡α²+ω_(p) ²−ω₀ ²>0 if α²+ω_(p) ²>ω₀ ² and Q(α,ω₀)=2ω₀α>0if α>0 and ω₀>0, then

$\begin{matrix}{{E_{x}\left( {t,z} \right)} = {D\;{\mathbb{e}}^{{\alpha\; t} - {{z{({P^{2} + Q^{2}})}}^{1/4}{\cos(\frac{\tan^{- 1}{({Q/P})}}{2})}}}{\sin\left\lbrack {{\omega_{0}t} - {{z\left( {P^{2} + Q^{2}} \right)}^{1/4}{\sin\left( \frac{\tan^{- 1}\left( {Q/P} \right)}{2} \right)}}} \right\rbrack}}} & {{Eq}.\mspace{14mu} 19}\end{matrix}$Thus, the envelope of the carrier is

$\begin{matrix}{D\;{\mathbb{e}}^{{\lbrack{{({P^{2} + Q^{2}})}^{1/4}{\cos(\frac{\tan^{- 1}{({Q/P})}}{2})}}\rbrack} \times {\{{{{({\alpha\; t})}/{\lbrack{{({P^{2} + Q^{2}})}^{1/4}\cos{(\frac{\tan^{- 1}{({Q/P})}}{2})}}\rbrack}} - z}\}}}} & {{Eq}.\mspace{14mu} 20}\end{matrix}$which has the form of a function F(v_(z)t−z) which propagates with acontrollable velocity v_(z) and controllable attenuation of a truncatedSD envelope without changing shape^([12]). The propagation velocity ofthe envelope is:

$\begin{matrix}{v_{z} = \frac{\alpha}{\left( {P^{2} + Q^{2}} \right)^{1/4}{\cos\left( \frac{\tan^{- 1}\left( {Q/P} \right)}{2} \right)}}} & {{Eq}.\mspace{14mu} 21}\end{matrix}$Note a similar results hold for {dot over (H)}_(y) in Eq. 15.

It is noted that the types of media discussed above (e.g., RLCtransmission lines and ionospheres) are merely examples. A waveformwhich may include a speedy delivery signal envelope modulated with asinusoidal signal may travel on other types of media without distortionsuch as, but not limited to travel of any energy wave including a signalenvelope traveling on a lossy dispersive media.

Example 11 Temperature Effects on SD Signal Delay in a Media

The temperature of a cable can influence the electrical SD signal delayin a media, especially metal wire line cables, such as RG-58/U coaxialcable. The effect can be predicted by the SD signal delay theory forwire line cable where the relation between the delay of a cable delayand a temperature of a cable can be useful in creating a new type ofprecision temperature measuring instrument.

1. Temperature Measurements

An RLC transmission line model can be used to model a metal line wire,e.g., a RG-58/U cable, at signal frequencies where the dielectric lossin the cable is small. The SD signal delay per unit length,(δ_(SD)=1/velocity) of an RLC transmission line isδ_(SD)=√{square root over (L(T)C(T))}{square root over(L(T)C(T))}√{square root over (1+τ/(L(T)/R(T)))}{square root over(1+τ/(L(T)/R(T)))}  Eq. 22where the SD signal input voltage applied to the cable is De^(αt) and

$\tau = {\frac{1}{\alpha}.}$The coaxial cable transmission line parameter, R(T), in Eq. 22 is thecombined resistance per unit length of the center conductor and theshield of the metal line wire. As such, the resistance of the coaxialcenter conductor and shield decreases with increasing temperature. Theinductance and capacitance can also change with temperature due to thechanges in the cross sectional dimensions of the cable due to thetemperature coefficient of expansion of the insulation(PE-Polyethylene). Therefore, the length of the conductor can alsoincrease slightly with increasing temperature (17 ppm/C.°) resulting ina small increase in the two way travel time of the TDR SD test pulse.

2. Calibration Mode

If the cable temperature can be controlled from one isothermaltemperature to another and delay measurements performed on the cable ateach temperature, then the calibrated relation between temperature andcable delay can be used to monitor the unknown cable temperature byperiodically measuring the cable delay. In one example, time domainreflectometer delay measurements of the two way travel of SD pulses wereperformed on an un-terminated 300 ft. RG-58/U cable while the cable washeld at a series of measured constant temperatures. Un-terminated, asdefined herein, is a media terminated by any impedance other than thecharacteristics impedance of the media. Referring to FIG. 54, theresults of different temperatures versus cable delay calibrationmeasurements are shown. These measurements were made over a period ofsix days at various isothermal temperatures from room temperature,approximately 22°, to freezing, approximately 0° C. Referring to FIG.55, a graph showing the average daily temperature plotted verses theaverage delay for each day. As seen from the results, the measurement ofSD signal delay is correlated with the cable temperature.

The inductance per unit length of coaxial cable is

$L = {\frac{\mu}{2\pi}{\ln\left( \frac{b(T)}{a} \right)}}$and the capacitance per unit length is

$C = {\frac{2{\pi ɛ}}{\ln\left( \frac{b(T)}{a} \right)}.}$Thus, √{square root over (LC(T))}=√{square root over (μ∈(T))}. Theresulting SD signal delay (e.g., the time of flight, TOF) duringpropagation from cable input to un-terminated cable end and reflectionback to the cable input is a2·l(T)·√{square root over (μ∈(T))}√{square root over(1+τ(L(T)/R(T)))}{square root over (1+τ(L(T)/R(T)))}.  Eq. 23The two measurements in FIG. 55 at room temperature with an 21.5° C.average together with two measurements near freezing with an −2.6° C.average will be used to estimate the change in the cable dielectricconstant over these range of temperatures.Assuming the length of the cable is 600 ft, the increase in delay due tothe thermal expansion (17 ppm/° C.) is 382 psec for a 24° C. increase intemperature.

The RG 58/U cable parameters are R(20° C.)=15.5 mΩ/ft, L(20° C.)=0.0801μH/ft, C(20° C.)=28.5 pF/ft. The coefficient of thermal expansion of thepolyethylene dielectric is 1,500-3,000×10⁻⁷/° C. Follow from Eq. 23 andthe above parameters,

$\begin{matrix}{{\sqrt{{\mu ɛ}\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)} \times {l\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)} \times \sqrt{\frac{1 + {\tau\left( \frac{R\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)}{L\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)} \right)}}{\sqrt{1 + {\tau\left( \frac{L\left( {21.5{^\circ}\mspace{14mu}{C.}} \right)}{R\left( {21.5{^\circ}\mspace{14mu}{C.}} \right)} \right)}}}}} = {\frac{{\mu ɛ}\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)}{\sqrt{{\mu ɛ}\left( {21.5{^\circ}\mspace{14mu}{C.}} \right)}} \times 0.9995920 \times 1.000498}} & {{Eq}.\mspace{14mu} 24}\end{matrix}$The measurement signal delay ratio is

$\frac{{{Signal\_ Delay}@{- 2.6}}{^\circ}\mspace{14mu}{C.}}{{{Signal\_ Delay}@21.5}{^\circ}\mspace{14mu}{C.}} = \frac{946.214}{945.842}$Thus,

${\frac{\sqrt{{\mu ɛ}\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)}}{\sqrt{{\mu ɛ}\left( {21.5{^\circ}\mspace{14mu}{C.}} \right)}} \times 1.000090} = \frac{946.214({nsec})}{945.842({nsec})}$which equates to

$\begin{matrix}{\frac{ɛ\left( {{- 2.6}{^\circ}\mspace{14mu}{C.}} \right)}{ɛ\left( {21.5\;{^\circ}\mspace{14mu}{C.}} \right)} = {\left( {\frac{1}{1.000090} \times \frac{946.214({nsec})}{945.842({nsec})}} \right)^{2} = 1.0006}} & {{Eq}.\mspace{14mu} 25}\end{matrix}$indicating the increasing the dielectric constant when the cable iscooled.

As the above equations show, if the cable length is accurately known ata given temperature (e.g. room temperature), and the nominal dimensionsof the cable as well as the metal and dielectric coefficients ofexpansion are also known, then the relation between SD signal cabledelay and cable temperature may be used to estimate the cable dielectricconstant at various temperatures.

3. Non-Negligible Dielectric Loss

In general cases, signals with higher signal frequency and dielectricloss in the cable are non-negligible. As such, when all four of theprincipal transmission line parameters are dependant on frequency aswell as temperature, T, then the SD signal propagation velocity becomes

$\begin{matrix}{v_{SD} = \frac{(\alpha)}{\sqrt{{{\overset{\_}{L}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}\alpha^{2}} + {\begin{pmatrix}{{{\overset{\_}{G}\left( {\alpha,T} \right)}{\overset{\_}{L}\left( {\alpha,T} \right)}} +} \\{{\overset{\_}{R}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}}\end{pmatrix}\alpha} + {{\overset{->}{R}\left( {{\alpha\; T},} \right)}{\overset{\_}{G}\left( {\alpha,T} \right)}}}}} & {{Eq}.\mspace{14mu} 26}\end{matrix}$where L(s,T), C(s,T), R(s,T), and G(s,T) are the Laplace transforms ofthe four line parameters. The cable delay per unit length, δ_(SD) isequal to 1/v_(SD), indicating the SD signal delay dependence ontemperature is

$\begin{matrix}{\delta_{SD} = {\left( \frac{1}{\alpha} \right) \cdot \sqrt{{{\overset{\_}{L}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}\alpha^{2}} + {\begin{pmatrix}{{{\overset{\_}{G}\left( {\alpha\;,T} \right)}{\overset{\_}{L}\left( {\alpha,T} \right)}} +} \\{{\overset{\_}{R}\left( {\alpha,T} \right)}{\overset{\_}{C}\left( {\alpha,T} \right)}}\end{pmatrix}\alpha} + {{\overset{\_}{R}\left( {\alpha,T} \right)}{\overset{\_}{G}\left( {\alpha,T} \right)}}}}} & {{Eq}.\mspace{14mu} 27}\end{matrix}$Metal cable temperature measuring instruments that analyze reflectedelectrical signal delay in a cable to estimate the temperature profilealong the length of the cable are feasible.

4. Thermometer

In one embodiment, an apparatus may be used to measure the temperatureof the media. Referring to FIG. 54, a time domain reflectometer (TDR)1000 includes a signal generator 1001 and a processor 1002 coupled tothe signal generator 1001. It is noted that signal generator 1001 andprocessor 1002 may be separate and distinct components. Alternatively,signal generator 1001 and processor 1002 may be an integral unit. Thesignal generator may generate a exponential waveform, such as a SDsignal, and may transmit the signal on a media (e.g., an interconnect,metal wire cable, etc.). The processor 1002 may determine the delay asdescribed in Eq. 22 of the generated waveform over a length of themedia. Based on the delay of the signal, the processor may be configureto determine the temperature of the media.

It is noted in the above examples that the delay is inverselyproportional to the temperature. However, it will be apparent to thoseskilled in the art that depending on the characteristics of the media,for example the dielectric constants, the delay may be proportional tothe temperature.

Example 12 Designing on Chip Interconnects with Reduced Delay

The delay of interconnects on high performance chips have been estimatedfrom lossy transmission line theory using an approximation which isbased on a distributed RC model of the transmission line which, untilrecently, have neglected inductive effects. New methods^([14]) nowemploy an RLC transmission line approximation for interconnects becauseinductive affects can no longer be neglected in integrated circuitsdesigned to operate with high speed clocks (e.g., gigahertz clocks).Using the SD signal in these interconnects permits the employment ofeither the RC or RLC line models within the same theoretical frameworkfor accurately analyzing circuit delay. For example, the RLCtransmission model using the telegrapher's equation as described in Eq.1 of Example 9 can be obtained by setting G to 0 which yields:

$\begin{matrix}{{{{LC}\frac{\partial^{2}v}{\partial t^{2}}} + {{RC}\frac{\partial v}{\partial t}}} = \frac{\partial^{2}v}{\partial x^{2}}} & {{Eq}.\mspace{14mu} 28}\end{matrix}$The SD signal delay per unit interconnect length, (δ_(SD)=1/velocity,τ=1/α), isδ_(SD)=√{square root over (LC)}√{square root over (1+τ/(L/R))}  Eq. 29.Note that if the SD signal waveform parameter τ equals (L/R), then thesignal delay/unit distance traveled is √{square root over (2)} largerthan √{square root over (LC)} (the minimum delay/unit length of theideal no loss line). And if τ is reduced to approximately (0.1)(L/R),then the SD signal delay/unit length is also reduced becoming only about5% smaller than √{square root over (LC)}.

Reducing τ to obtain smaller line delay with the SD signal, is alsoaccompanied by an increase in the propagation attenuation vs. travellength of the truncated SD signal. However, the attenuation of thistruncated signal on an RLC line at length l is bounded, being no worsethan a limiting maximum value of

${\mathbb{e}}^{{- {({\frac{R}{2}\sqrt{C/L}})}}l}$as τ→0.

1. SD Repeater Insertion

Repeater insertion is generally required in long lossy on chipinterconnect lines^([15]) because of signal attenuation. As such, SDrepeater insertion methods can be employed in long lines to reduceinterconnect delay while maintaining signal amplitude constraints foradequate input signal integrity at the repeaters. In one embodiment,repeaters having a SD waveform output can be implemented in CMOS wherethe high speed latched comparator circuits in CMOS contain a middlestage (unstable feedback loop) that can produce the positive exponentialwaveform^([16] [17]).

A repeater insertion design for a 1 cm line using line parametersR=40.74 Ohm/mm, L=1.52 nH/mm, and C=228.5 fF/mm, and considering atarget clock frequency of 10 GHz for a 0.07 μm CMOS technology isdemonstrated. First, a cadence simulation of a 0.18 μm CMOS processtechnology of a re-settable latch with inverters as output buffers wasperformed and yielded a SD signal output with τ=60 ps. For the 0.07 μmCMOS technology, the estimated SD signal is scaled such that τ isapproximately 23 ps. For a 10 GHz clock, τ can be assumed to be reducedby a factor 10 (τ is approximately 3 ps). FIG. 57 shows the line delayversus τ. For τ=3 ps, the results in a line delay/unit length isapproximately 19.4 ps/mm. FIG. 58 illustrates the truncated SD signalattenuation as a function of line length.

Choosing a maximum allowed attenuation of approximately 0.7 for eachline segment length allows a maximum of 1.46 mm per segment. Assumingsix segments over the total 10 mm length (the total 10 mm wire delay,excluding the repeater delays is 10×19.4 ps/mm=194 ps/mm), and arepeater delay, T_(R), of 3τ to 6τ. In this example, the repeater delayis approximately 10 ps which yields a total delay of the segmented lineequal to 264 ps ((7×10 ps)+194 ps). This is approximately three clockperiods for a 10 GHz clock. The minimum delay of an un-segmented (norepeaters) ideal no loss line of 10 mm length (10 mm×18.6 ps/mm) is 186ps or approximately two clock periods at 10 GHz. The ratio (%) of the noloss 10 mm line delay (186 ps) to the total delay of SD repeatersolution for the lossy 10 mm line (264 ps) is 70%.

By using the SD repeater insertion, any performance limitinginterconnect delay larger than l√{square root over (LC)} over long pathsof length l can be reduced by making the repeater delay and parameter τof the SD repeater output signal smaller.

Example 13 Encoding Propagation Properties into the Signal Waveforms

As noted above, the SD waveform and accompanying propagation propertiescan be incorporated into the signal waveforms of digital andcommunication systems. In communication systems, the coding modalitiesunique to the SD waveform can be utilized that complement currentcommunication waveform coding techniques. For example, since the SDwaveform does not change shape during propagation in dispersive andlossy media, the SD shape parameter α (exponential coefficient) maybevaried from one transmitted SD pulse to another with the pattern ofchange in a detected at the receiver. In this manner, the value of α maybe coded to convey information transmission in the channel.

Additionally, the process of encoding also allows multiple distinctvalues of α to be simultaneously incorporated into SD portions of theleading edge of a pulse. An example is illustrated in the figures thatfollow. FIG. 59 shows a graphical user interface that demonstrates aninput pulse with two distinct shapes of SD signals comprising theleading edge of the pulse. The lower voltage range consists of an SDsignal with a shape parameter α₁ which has a value of 2×10⁷ (l/sec) andthe higher voltage range of the leading edge consists of a second SDwaveform shape with α₂ which has a value of 3×10⁷ (l/sec). Thesedistinct values are incorporated in the leading edge of the pulse andare preserved during the propagation in a 600 ft in a RG 58/U coaxialcable. FIG. 60 illustrates the preservation of the first SD signalsection of the pulse with shape parameter α₁ with the input pulse on theleft of the figure and the reflected pulse after traveling 600 ft on theright. FIG. 61 illustrates the natural log of the voltage amplitudes ofthese input and reflected signals with slopes of both consistent withα₁. FIG. 62 illustrates the output of ratio of filters used on thesignal of FIG. 60. and FIG. 63A illustrate the shape preservation of thesecond SD signal portion of the leading edge with α₂, where FIG. 63Billustrate natural log of the voltage amplitudes of these input andreflected signals with slopes of both consistent with α₂. FIG. 63Cillustrates the output of ratio of filters used on the signal of FIG.63A. FIG. 64 shows a graphical user interface that demonstrates a pulsewith these two sections of SD signals interchanged with α₁ for the firstsection and α₂ for the second section. FIG. 65 shows the preservation ofthe shapes for the α₁ with the input pulse on the left of the figure andthe reflected pulse after traveling 600 ft on the right. FIG. 66illustrates the natural log of the voltage amplitudes of these input andreflected signals with slopes of both consistent with oil. FIG. 67illustrates the output of ratio of filters used on the signal of FIG.65. Similarly, FIGS. 68A, 68B, and 68C show the preservation of theshapes for the α₂ with the input pulse on the left of the figure and thereflected pulse after traveling 600 ft on the right.

These two pulses with composite SD shapes in their leading edge,together with two pulses with distinct single SD shapes corresponding toeither α₁ or α₂ in their leading edge, together provide a modulationscheme capable of conveying two bits information in the channel. Thestandard pulse amplitude modulation process may be combined with thismodulation of SD shapes creating a composite modulation technique toenhance data rate by two bits per symbol period with essentially noincrease in bandwidth of the transmitted symbols. The values of α in theSD sections of these pulses may be detected (e.g., decoded) as discussedin Example 7.

Example 14 Bias Adjustments to Decrease the Standard Deviation of SignalDelay Measurement

The leading edge of a SD signal, in particular, a truncated SD signal,that is transmitted on a media has an exponential shape. However, thetransmission of this signal may have a slow transient condition. Thetransient condition may be a bias resulting from a long, slow, decayingtail. This bias may be added to the exponential waveform, and thus mayskew the waveform. This example demonstrates an algorithm for removingthe bias.

1. Algorithm A: Compensation Algorithm for a Known or an Unknown a Valueof Applied Signal

Referring to FIG. 69, a measured waveform before any bias is shown. Thereceived 10,000 discrete data, from 0 nsec to 2,00 nsec will first beseparated into two categories: applied signal including the first 3,500discrete data from 0 nsec to 700 nsec and reflected signal including theremaining data from 700 nsec to 2,000 nsec, as shown in FIG. 69. The SDthreshold overlap region is defined as the signal voltage thresholdrange between 25% and 75% of the peak voltage of the reflected signal.The slope of the natural log of the applied signal and the reflectedsignal in the overlap region needs to be adjusted to match the α valuefor the applied signal, which is 2×10⁷ l/sec, as shown in FIG. 70.

A. Compensation Algorithm for a Known α Value of Applied Signal

Referring to FIG. 71A, the signal shown in the solid line represents ameasured exponential waveform with an added unknown positive bias, +B.For this example, +B is approximately +0.05 V. It is noted that the biasvoltage may be higher. Alternatively, the bias voltage may be lower. Thesignal shown in the dotted line is the measured exponential waveformwith bias B removed. In FIG. 71B, the natural log of the measuredexponential waveform with bias is shown in the solid line and thenatural log of the exponential waveform without bias +B is shown withthe dotted line. As seen in FIG. 71B, the estimated slope of the naturallog plot of the measured signal with positive bias +B in the overlapregion (solid line) will be smaller than α. On the other hand, theestimated slope of the natural log plot, in the overlap region of themeasured signal with the bias +B removed (dotted line) will have a slopesubstantially equal to α (2×10⁷ (l/sec)).

Referring to FIG. 72A, the signal shown in the solid line is a measuredexponential waveform with an added unknown negative bias, −B. For thisexample, −B is approximately −0.05 V. It is noted that the bias voltagemay be higher. Alternatively, the bias voltage may be lower. The signalshown with a dotted line is the exponential signal without negative bias−B. In FIG. 72B, the natural log of the exponential waveform withnegative bias −B is shown in the solid line and the natural log of theexponential waveform without bias −B is shown with the dotted line. Itrepresents that the slope of the natural log plot of the measured signalwith negative bias −B in the overlap region (solid line) will be greaterthan α. Therefore, after removing the negative bias −B, the natural logplot in the overlap region, of the measured signal will have a slopesubstantially equal to α.

As stated above, the unknown bias must be removed in order to obtainaccurate threshold time of flight, TOF, values. In one embodiment, theunknown bias B can reflected in the slope of the natural log plot of theexponential waveform plus bias in the SD threshold overlap region. Assuch, the unknown bias can be estimated and removed. In this biasremoval, the error tolerance of residual bias of both the applied andreflected signals is chosen to be 1 μV.

i. Applied Signal

In this experiment, the applied signal is be a pulled-down signal with avoltage value such that the measured signal graph becomes that of anexponential waveform with an unknown negative bias. In one embodiment,the voltage of the applied signal may be approximately 0.01 V. First,the graph is lifted gradually with an added incremental voltage valuewhich is initially chosen to be +0.001 volts at the beginning (0.01/10volts) and is incremented until the slope of natural log of the signalversus the time is lower than the known α value of the applied signal.Next, the process is continued by adding one incremental negative biasvoltage level (−0.001 V). Then the previous process of incrementallyadding positive voltage level (+0.001/10 volts) +0.0001 V is performeduntil the natural log plot has a positive bias again as indicated by theslope of the log plot of the SD threshold region being less than α. Atthis point, the waveform becomes an exponential with a positive additivevoltage bias. The previous process is again repeated by subtracting+0.0001 V from the waveform resulting a negative bias again. Thepositive incremental bias level 0.00001 V is added to the signalrepeating the process describe above. Then the whole process repeatedone more time with a new incremental voltage level, e.g., 1 μV(+0.00001/10 volts). At this point the bias adjustment process isconcluded. This process is separately used to reduce the residual biasto less than 1 μV for both the applied and reflected signals. Thenatural log of both signals with the bias less than 1 μV versus timeplot is shown in FIG. 70. The total shifted voltage of applied andreflected signal is named as Bias₁ and Bias₂, respectively and isillustrated in FIG. 73.

Once the biases of both signals have been reduced, the final resultsobtained in this manner will be used to calculate the constant thresholdTOF values in each of a series of incremental threshold voltages.

B. Compensation for Unknown α Value of Applied Signal

Referring to FIGS. 74A and 74B and FIGS. 75A and 75B, the SD thresholdoverlap region of a measured applied signal or a signal is separatedinto a SD threshold lower overlap region and a SD threshold higheroverlap region, in which the average threshold voltage level determiningthe border between higher and lower overlap region. In FIG. 74A, thesignal represented by the solid line is an exponential waveform with anadded positive bias +B. In this example, the positive bias +B is 0.05 V.The signal represented by the dotted line is an exponential waveformwithout any bias. The natural log of the exponential waveform with bias(solid line) and the natural log of the exponential waveform withoutbias (dotted line) are shown in FIG. 74B. The slope in the SD thresholdlower region is approximately 1.62×10⁷ (l/sec) where the slope in the SDthreshold higher region is approximately 1.77×10⁷ (l/sec). These valuesshow that the slope of natural log plot is increased from the SDthreshold lower region to the SD threshold higher region when theunknown bias is positive. Similarly, referring to FIG. 75A, the signalshown in solid line is the exponential waveform with an added negativebias −B. In this example, −B is a −0.05 volts. The signal shown indotted line is the exponential waveform without bias. As shown in FIG.75B, the natural log of the exponential waveform without the bias −B isshown in a dotted line and the natural log of the exponential waveformwithout the bias −B is shown in solid line. The slope of the SDthreshold lower region and the slope of the SD threshold higher regionare estimated as 2.61×10⁷ (l/sec) and 2.32×10⁷ (l/sec), respectively. Asa result, decreasing the slope of natural log plot from the lower regionto higher region implies the bias is negative. Therefore, from thecorrelation of bias and the difference of slope in two regions (i.e.,the SD threshold higher region and SD threshold lower region), the biascan be estimated and removed.

2. Algorithm B: Compensating Algorithm Using the Difference Between theAdjusted Applied Signal and the Reflected Signal

For the algorithm described above (e.g., algorithm for a known or anunknown α value of applied signal), adjusts both the applied and thereflected signals, which are termed as Bias1 and Bias2, respectively. Acalibration program can be applied to a series of measured constantthreshold TOF values for both Bias1 and Bias2 on the same length of themedia, as shown in FIG. 73. The variation patterns of Bias1 and Bias2are observed to be similar. Interestingly enough, further comparison ofBias1 and Bias2 reveals that another similarity exists between measuredconstant threshold TDR TOF measurements and the difference between Bias₁and Bias₂, which is shown in FIG. 74A. This phenomenon implies that thedifference of bias (ΔBias=Bias1−Bias2), illustrated in FIG. 74Bcorrelates with the measured constant threshold TDR TOF measurements inFIG. 74A.

In this example, the standard deviation of measured constant thresholdTDR TOF values can be drastically reduced. For example, let ΔB ( ΔB=Bias1 − Bias2 ) denote the difference of the means of Bias1 and Bias2,where Bias1 and Bias2 can be determined from the measured applied waveand the reflected wave, respectively using the algorithm for a known oran unknown α value of applied signal described above. ΔB can begenerated from calibration program used to partially adjust the bias forthe reflected signal. For this example, ΔB is found to be approximately0.00591 V. This algorithm described herein as the “B algorithm”, usesthe bias of applied signal, PBias1, calculated by using the algorithmdescribed above, described herein as the “A algorithm” (i.e., algorithmfor a known or an unknown α value of applied signal). The bias ofreflected signal, PBias2, is ΔB subtracted from PBias1 (PBias2=PBias1−ΔB). Next, PBias1 and PBias2 will be added to the applied and reflectedsignal waveform respectively to yield the bias compensated waveforms.Accordingly, the time values corresponding to 100 individual voltagesthreshold voltage crossing times in the threshold overlap region arecalculated and averaged. The mean value of these 100 threshold crossingTOF in reaching 100 threshold voltages is regarded as the TOF value inthis measurement. By so doing, the new results of 60 measured constantthreshold TDR TOF measurements are shown in FIG. 75, and associatedhistogram is shown in FIG. 76. As seen in FIG. 75, the range of newmeasured constant threshold TDR TOF measurements is approximately 45nsec, and the standard deviation of measured constant threshold TDR TOFis approximately 9.7 psec.

Comparison of the measured constant threshold TDR TOF values andhistograms of employing the A algorithm and the B algorithm (FIGS. 75and 76) shows that the range of measured constant threshold TDR TOF in60 measurements is obviously reduced from 131 psec to 45 psec. Also thestandard deviation of measured constant threshold TDR TOF is reducedfrom 22.7 psec to 9.7 psec. Moreover, all measured constant thresholdTDR TOF measurements obtained from the B algorithm bias adjustment arelying within the range obtained from A algorithm. As such, the Balgorithm has more accurate measured constant threshold TDR TOFestimates with less than approximately half of the standard deviationscompare with the A algorithm.

All of the methods and apparatuses disclosed and claimed can be made andexecuted without undue experimentation in light of the presentdisclosure. While the apparatus and methods of this invention have beendescribed in terms of embodiments, it will be apparent to those of skillin the art that variations may be applied to the methods and in thesteps or in the sequence of steps of the method described withoutdeparting from the concept, spirit and scope of the invention. Inaddition, modifications may be made to the disclosed apparatus andcomponents may be eliminated or substituted for the components describedwhere the same or similar results would be achieved. All such similarsubstitutes and modifications apparent to those skilled in the art aredeemed to be within the spirit, scope and concept of the invention asdefined by the appended claims.

The terms a or an, as used herein, are defined as one or more than one.The term plurality, as used herein, is defined as two or more than two.The term another, as used herein, is defined as at least a second ormore. The terms including and/or having, as used herein, are defined ascomprising (i.e., open language). The term coupled, as used herein, isdefined as connected, although not necessarily directly, and notnecessarily mechanically. The term approximately, as used herein, isdefined as at least close to a given value (e.g., preferably within 10%of, more preferably within 1% of, and most preferably within 0.1% of).The term substantially, as used herein, is defined as at leastapproaching a given state (e.g., preferably within 10% of, morepreferably within 1% of, and most preferably within 0.1% of). The termmeans, as used herein, is defined as hardware, firmware and/or softwarefor achieving a result. The term program or phrase computer program, asused herein, is defined as a sequence of instructions designed forexecution on a computer system. A program, or computer program, mayinclude a subroutine, a function, a procedure, an object method, anobject implementation, an executable application, an applet, a servlet,a source code, an object code, a shared library/dynamic load libraryand/or other sequence of instructions designed for execution on acomputer system.

The appended claims are not to be interpreted as includingmeans-plus-function limitations, unless such a limitation is explicitlyrecited in a given claim using the phrase(s) “means for” and/or “stepfor”. Subgeneric embodiments of the invention are delineated by theappended independent claims and their equivalents. Specific embodimentsof the invention are differentiated by the appended dependent claims andtheir equivalents.

REFERENCES

-   [1] Gruodis and Chang, IBM J. Res. Develop., 25:25-41, 1981.-   [2] Sauter, In: Nonlinear Optics, John Wiley & Sons Inc., NY, 127,    1996.-   [3] Agrawal, In: Nonlinear Fiber Optics, Academic Press, Sand Diego,    3^(rd) Ed., 127, 2001.-   [4] Jackson, In: Fourier Series and Orthogonal Polynomials, George    Banta Co., Inc., Menasha, 156-157, 1957.-   [5] Szabo and Ostlund, In: Modern Quantum Chemistry, Dover Pub.    Inc., Mineola, 142-145, 1989.-   [6] Proakis, In: Digital Communications, WCB McGraw-Hill, 3^(rd)    Ed., Boston, 167-173, 1995.-   [7] T1.417-2001 American National Standard-Spectrum Management for    Loop Transmission Systems, Annex B: Loop Information, p. 84.-   [8] Deutsch et. al., IBM J. Res. Develop., 34(4), 605, 1990.-   [9] Wheeler and Crummeft, Am. J. Phys., 55(1):33-37, 1987.-   [10] Sommerfeld, In: Mechanics of Deformable Bodies Lectures on    Theoretical Physics, Academic Press, Inc., NY, Vol. 1:96-98, 1950.-   [11] Weber, In: Linear Transcient Analysis, Wiley, NY, Vol. II:273,    1954.-   [12] Flake and Bishop, In: Signal Propagation without Distortion in    Dispersive Lossy Media, 11^(th) IEEE Intl. Conf. Electronics,    Circuits and System Proceed., Israel, 2004.-   [13] Kalluri, In: Electromagnetics of Complex Media, CRC Press,    1998.-   [14] Ismail and Friedman, IEEE Trans. Circuits Syst. II, 48:471-481,    2001.-   [15] Davidson et al., IEEE Trans. Comp. Packag. Manufact. Technol.,    20(4), 1997.-   [16] Razavi and Wooley, IEEE J. Solid-state Circuits, 27:1916-1992,    1992.-   [17] Johns and Martin, In: Analog Integrated Circuit Design, Wiley &    Sons, Inc., NY, 317-326, 1997.

1. A method comprising: determining a length of a lossy electrical cable using a time-domain reflectometry (TDR) technique, wherein said determining the length of the lossy electrical cable using the TDR technique includes: sending an electrical pulse into the lossy electrical cable, wherein the electrical pulse comprises a leading edge portion and a non-leading edge portion, wherein the leading edge portion comprises a positive exponential waveform, wherein the positive exponential waveform maintains its shape during transmission through the lossy electrical cable, wherein the non-leading edge portion does not maintain its shape during transmission through the lossy electrical cable; and detecting a reflected signal in response to the electrical pulse, wherein the reflected signal is due to presence of an impedance discontinuity at an end of the lossy electrical cable, wherein the reflected signal comprises a leading-edge portion and a non-leading edge portion, wherein the leading-edge portion of the reflected signal comprises the positive exponential waveform; determining a time of flight between the positive exponential waveform within the leading edge portion of the electrical pulse and the positive exponential waveform within the leading edge portion of the reflected signal; determining the length of the lossy electrical cable based on the time of flight.
 2. The method of claim 1, wherein the positive exponential waveform within the leading-edge portion of the electrical pulse conforms to an exponential function Ae^(αt), wherein A is a constant, wherein α is a positive constant, wherein t is time.
 3. The method of claim 1, wherein the leading-edge portion of the reflected signal includes a non-exponential component in addition to said positive exponential waveform, wherein the positive exponential waveform within the leading edge portion of the electrical pulse lasts for a time duration sufficient to ensure that the positive exponential waveform within the leading edge portion of the reflected signal dominates the non-exponential component of the leading edge portion of the reflected signal.
 4. The method of claim 1, wherein said determining a time-of-flight includes: determining a time t₁ when the positive exponential waveform within the leading edge portion of the electrical pulse crosses a threshold signal level; determining a time t₂ when the positive exponential waveform within the leading edge portion of the reflected signal crosses the threshold signal level; and computing a difference between t₂ and t₁.
 5. The method of claim 4, wherein the threshold signal level is between lower bound L and upper bound U, wherein L is approximately 0.25 P, wherein U is approximately 0.75 P, wherein P is the peak amplitude of the reflected signal.
 6. The method of claim 1, wherein said determining the length of the lossy electrical cable using the TDR technique also includes: capturing the electrical pulse and the reflected signal into a memory buffer; repeatedly scanning the memory buffer to determine corresponding time-of-flight values; and averaging the time-of-flight values; wherein each of said scans of the memory buffer includes: determining a time t₁ when the positive exponential waveform within the leading edge portion of the captured electrical pulse crosses a threshold signal level; determining a time t₂ when the positive exponential waveform within the leading edge portion of the captured reflected signal crosses the threshold signal level; and computing a difference between t₂ and t₁; wherein each of said scans uses a different value of the threshold signal level.
 7. A system comprising: a signal generator that stimulates a lossy transmission medium with an input pulse, wherein the input pulse includes a leading-edge portion and a non-leading edge portion, wherein the leading edge portion comprises a positive exponential waveform, wherein the positive exponential waveform maintains its shape during transmission through the lossy transmission medium, wherein the non-leading edge portion of the input pulse does not maintain its shape during transmission through the lossy transmission medium; and a receiver configured to: (a) capture a response pulse from the lossy transmission medium, wherein the response pulse includes a leading-edge portion and non-leading edge portion, wherein the leading edge portion of the response pulse comprises the positive exponential waveform; (b) determine a first time-of-flight between the positive exponential waveform within the leading edge portion of the input pulse and the positive exponential waveform within the leading edge portion of the response pulse; and (c) compute a length of the lossy transmission medium using the first time-of-flight.
 8. The system of claim 7, wherein the receiver is configured to perform (b) by: determining a time t₁ when the positive exponential waveform within the leading edge portion of the input pulse crosses a threshold signal level; determining a time t₂ when the positive exponential waveform within the leading edge portion of the response pulse crosses the threshold signal level; and computing a difference between t₂ and t₁.
 9. The system of claim 8, wherein the threshold signal level is between lower bound L and upper bound U, wherein L is approximately 0.25 P, wherein U is approximately 0.75 P, wherein P is the peak amplitude of the response waveform.
 10. The system of claim 7, wherein the receiver is configured to capture the input pulse and the response pulse into a data buffer, wherein said computing the length of the lossy transmission medium includes: computing a number of time-of-flight values; and computing the first time-of-flight by averaging the time-of-flight values; wherein each of said time-of-flight values is computed by: determining a time t₁ when the positive exponential waveform within the leading edge portion of the captured input pulse crosses a threshold signal level; determining a time t₂ when the positive exponential waveform within the leading edge portion of the captured response pulse crosses the threshold signal level; and computing a difference between t₂ and t₁; wherein the computation of each of the time-of-flight values uses a different value of the threshold signal level.
 11. The system of claim 7 further comprising a T-connector and one or more of the following: an oscilloscope and a display device.
 12. The system of claim 7, wherein the lossy transmission medium is a lossy electrical cable.
 13. The system of claim 7, wherein the lossy transmission medium is a lossy fiber optic cable.
 14. The system of claim 7, wherein the lossy transmission medium is an acoustic medium, wherein the input pulse is an acoustic pulse.
 15. A method comprising: sending an electrical pulse into a lossy transmission medium, wherein the electrical pulse comprises a leading edge portion and a non-leading edge portion, wherein the leading edge portion comprises a positive exponential waveform, wherein the positive exponential waveform maintains its shape during transmission through the lossy transmission medium, wherein the non-leading edge portion does not maintain its shape during transmission through the lossy transmission medium; and detecting a response signal from the lossy transmission medium, wherein the response signal is produced in response to the electrical pulse, wherein the response signal comprises a leading-edge portion and a non-leading edge portion, wherein the leading-edge portion of the response signal comprises the positive exponential waveform; determining a time of flight between the positive exponential waveform within the leading edge portion of the electrical pulse and the positive exponential waveform within the leading edge portion of the response signal.
 16. The method of claim 15, wherein said sending comprises sending the electrical pulse into the lossy transmission medium at a first point of the lossy transmission medium, wherein said detecting includes detecting the response signal from a second point of the lossy transmission medium, where the method further comprises: determining a distance of travel within the lossy transmission medium between the first point and the second point based on the time of flight.
 17. The method of claim 15, wherein said sending comprises sending the electrical pulse into the lossy transmission medium at a point of the lossy transmission medium, wherein said detecting includes detecting the response signal from the same point of the lossy transmission medium, wherein the response signal is due to a reflection from an impedance discontinuity in the lossy transmission medium, wherein the method further comprises: computing a distance within the lossy transmission medium from the point to the impedance discontinuity based on the time of flight.
 18. The method of claim 15, wherein said determining a time of flight includes: determining a time t₁ when the positive exponential waveform within the leading edge of the electrical pulse crosses a threshold signal level; determining a time t₂ when the positive exponential waveform within the leading edge of the response signal crosses the threshold signal level; and computing a difference between t₂ and t₁.
 19. The method of claim 15, wherein the lossy transmission medium is a lossy electrical cable.
 20. The method of claim 15, wherein the lossy transmission medium is a lossy interconnect. 